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Time Optimum Problems for Unbounded Differential Inclusion*
16th IFAC workshop on Control Optimization 16th IFAC workshop on Control Applications Applications of of Optimization 16th IFAC workshop on of October 2015. Garmisch-Partenkirchen, 16th IFAC6-9, workshop on Control Control Applications ApplicationsGermany of Optimization Optimization October 6-9, 2015. Garmisch-Partenkirchen, Germany Available online October 6-9, 2015. Garmisch-Partenkirchen, Germany October 6-9, 2015. Garmisch-Partenkirchen, Germany at www.sciencedirect.com
ScienceDirect IFAC-PapersOnLine 48-25 (2015) 150–155
Time Optimum Problems for Time Unbounded for Unbounded Time Optimum Optimum Problems Problems for Unbounded Differential Inclusion Differential Inclusion Differential Inclusion ∗ ∗ ∗ ∗
Moscow Moscow Moscow Moscow
∗ Evgenii S. Evgenii S. Polovinkin Polovinkin ∗∗∗ Evgenii Evgenii S. S. Polovinkin Polovinkin Institute of Physics and Technology, Moscow, Institute of Physics and Technology, Moscow, Institute of Physics and Technology, Moscow, Institute of Physics and Technology, Moscow, (e-mail: [email protected]). (e-mail: [email protected]). (e-mail: [email protected]). [email protected]). (e-mail:
Russia, Russia, Russia, Russia,
Abstract: Based on the properties of solutions of a differential inclusion with unbounded Abstract: of solutions of inclusion with Abstract: Based Based on on the the properties properties of side, solutions of aaa differential differential inclusion with unbounded unbounded Abstract: Based on the properties of solutions of differential inclusion with unbounded measurable-pseudo-Lipschitz right-hand the necessary conditions in time optimum problem measurable-pseudo-Lipschitz right-hand side, the necessary conditions in time optimum problem measurable-pseudo-Lipschitz right-hand side, the necessary conditions in time optimum problem measurable-pseudo-Lipschitz right-hand side, the necessary conditions in time optimum problem on set of solutions of the unbounded differential inclusion are obtained. on set of solutions of the unbounded differential inclusion are obtained. on set of solutions of the unbounded differential inclusion are obtained. on set of solutions of the unbounded differential inclusion are obtained. DOI: 10.1134/S0081543813080099 DOI: 10.1134/S0081543813080099 DOI: DOI: 10.1134/S0081543813080099 10.1134/S0081543813080099 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: differential differential inclusion, inclusion, necessary necessary optimality optimality conditions, conditions, time time optimum optimum problem. problem. Keywords: Keywords: differential differential inclusion, inclusion, necessary necessary optimality optimality conditions, conditions, time time optimum optimum problem. problem. 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION 1. 1. INTRODUCTION In the paper Aubin (1984) there was conIn the the paper paper Aubin Aubin (1984) (1984) there there was was introduced introduced aaa conconIn In the of paper Aubin (1984) multi-valued there was introduced introduced awhich condition pseudo-Lipschitz mapping, dition of pseudo-Lipschitz multi-valued mapping, which dition of pseudo-Lipschitz multi-valued mapping, which dition of pseudo-Lipschitz multi-valued mapping, which proved to be more convenient for research of unbounded proved to to be be more more convenient convenient for for research research of of unbounded unbounded proved proved to be mappings more convenient forLipschitz researchcondition. of unbounded multi-valued than the multi-valued mappings than the Lipschitz condition. multi-valued multi-valued mappings mappings than than the the Lipschitz Lipschitz condition. condition. Generalizations of J. -P. Aubin condition to the nonGeneralizations of J. -P. Aubin condition to the the nonnonGeneralizations of J. -P. Aubin condition to Generalizations of J. -P. Aubin condition to the nonautonomous case, i.e. case when the mapping F takes the autonomous case, i.e. case when the mapping F takes the autonomous case, case when mapping F the n i.e. autonomous case, i.e.P(R casennn ), when the mapping F takes takes and the n → P(R ), werethe first done Loewen Loewen and form F :: T × R n → were first done form F T × R n → P(Rn ), were first done Loewen and form F : T × R → P(R ), were first done Loewen and form F : T × R Rockafellar (1994) and later Ioffe (2006). In these works Rockafellar (1994) and later Ioffe (2006). In these works Rockafellar (1994) and later (2006). works Rockafellar (1994) and global later Ioffe Ioffe (2006). In In these these works some conditions of the pseudo-Lipschitz are given. some conditions of the global pseudo-Lipschitz are given. some conditions conditions of of the the global global pseudo-Lipschitz pseudo-Lipschitz are are given. given. some In this paper, we consider the differential inclusion with In this paper, we consider the differential inclusion with In this paper, we consider the differential inclusion with In this paper, we consider the differential inclusion with the other local conditions called conditions of measurablethe other local conditions called conditions of measurablethe other local conditions called conditions of measurablethe other local conditions called conditions of measurablepseudo-Lipschitz mappings, which first were proposed pseudo-Lipschitz mappings, which first were proposed pseudo-Lipschitz mappings, which first proposed pseudo-Lipschitz mappings, which first were were(2013)). proposed by the author (for example, see Polovinkin In by the author (for example, see Polovinkin (2013)). In by the author (for example, see Polovinkin (2013)). In by the author (for example, see Polovinkin (2013)). In these works, under the conditions of measurable-pseudothese works, under the conditions of measurable-pseudothese works, under the conditions of measurable-pseudothese works, under the conditions of measurable-pseudoLipschitz right-hand side of the differential inclusion many Lipschitz right-hand side of differential inclusion many Lipschitz right-hand side of the theexistence differential inclusion many Lipschitz right-hand side of the differential inclusion many results were received, namely: theorem for soluresults were received, namely: existence theorem for soluresults were received, namely: existence theorem for soluresults were received, namely: existence theorem for solutions, theorem on the relaxation of differential inclusion, tions, theorem on the relaxation of differential inclusion, tions, theorem on the relaxation of differential inclusion, tions, theorem on the relaxation of differential inclusion, theorem on the differentiation of the initial data and other theorem on the differentiation of the initial data and other theorem on the differentiation of data other theorem onof the differentiation of the the initial initial data and and other properties trajectorys of differential inclusion with unproperties of trajectorys of differential inclusion with unproperties of trajectorys of differential inclusion with unproperties of trajectorys of differential inclusion with unbounded right-hand side. bounded right-hand side. bounded right-hand side. bounded right-hand side. This work is devoted to the development of the results This work is to of results This work is devoted devoted to the the development development of the the results This work is devoted to the development of the results on the study of optimization problems with differential on the study of optimization problems with differential on the study of optimization problems with differential on the study of optimization problems with differential inclusions, obtained Polovinkin and Smirnov (1986a,b), inclusions, obtained Polovinkin and Smirnov (1986a,b), inclusions, obtained Polovinkin and Smirnov (1986a,b), inclusions, obtained (1991, Polovinkin and Smirnov (1986a,b), and later Polovinkin 1993, 1995, 2013, 2014), from and later Polovinkin (1991, 1993, 1995, 2013, 2014), from and later Polovinkin (1991, 1993, 1995, 2013, 2014), from and later Polovinkin (1991, 1993, 1995, 2013, 2014), from the case of the bounded right-hand side of the differential the case of the bounded right-hand side of the differential the case of the bounded right-hand side of the differential the case of the bounded right-hand side of the differential inclusion to to the the case case where where the the right right side side is is unbounded. unbounded. inclusion inclusion inclusion to to the the case case where where the the right right side side is is unbounded. unbounded. 2. MAIN MAIN NOTATIONS NOTATIONS AND AND DEFINITIONS DEFINITIONS 2. 2. 2. MAIN MAIN NOTATIONS NOTATIONS AND AND DEFINITIONS DEFINITIONS .. We denote line segment by T = [t Euclidean 0, t We denote line segment T = tt11 ]]] and and Euclidean .. n[t ..by We denote segment by T = [t00|,,, x and Euclidean nline 1 We denote segment by T = t ] and Euclidean nline n[t0 space by R . Let B (a) = {x ∈ R − a < r} and 1 . r n n space by R . Let B (a) = {x ∈ R | x − a r} and . r n . Let nBr (a) = {x ∈ Rn | x − a < space by R < r} and . space by R . Let B (a) = {x ∈ R | x − a < r} and . r n B (a) = {x ∈ R | x − a ≤ r}. The distance from . r (a) = n B {x ∈ R | x − a ≤ r}. The distance from . r (a) = {x ∈ Rnn | x − a ≤ r}. The B distance from n r B (a) = {x ∈ R | x − a ≤ r}. The from n to n distance the point x ∈ R the subset A ⊂ R we denote by r n the point subset A R denote by .. x n to the point x ∈ ∈ R R− to |the the subset A ⊂ ⊂ Rnn we wefunction denote for by the point x ∈ R to the subset A ⊂ R we denote by (x, A) = inf{x y y ∈ A}. The support . (x, A) = inf{x − y | y ∈ A}. The support function for . . (x, A) A) A =⊂ inf{x − y y | yydenote ∈ A}. A}. by Thes(p, support function for| n − .. function (x, = inf{x | ∈ The support for n we subset R will A) = sup{p, x n subset A ⊂ R will denote by s(p, A) = || . sup{p, x n we subset A ⊂ R we will by s(p, A) = x subset Ap ⊂ R will denote denote by s(p, A) R =nnn sup{p, sup{p, x | n . we x ∈ A}, ∈ A subset K in the space we will call n x ∈ A}, p ∈ R . A subset K in the space R we will call n . A subset K in the space Rn we will call x ∈ A}, p ∈ R x ∈ A}, p ∈ R . A subset K in the space R we will call This work was This work was This work was Research (project This work was Research (project Research (project (project Research
supported by the Russian supported by the Russian supported by the Russian nos. 13-01-00295a). supported by the Russian nos. 13-01-00295a). nos. 13-01-00295a). 13-01-00295a). nos.
Foundation Foundation Foundation Foundation
for for for for
Basic Basic Basic Basic
cone if for any x ∈K and any λ > 0 the inclusion λx ∈K cone if for x K any λ > inclusion λx cone if Polar for any anycone x∈ ∈ to K and and any λK > 000⊂the the inclusion λx ∈ ∈K K cone for any x ∈ K and > the λx ∈ K holds.if the any coneλ Rnnninclusion is the the following following holds. Polar cone to the cone K ⊂ R is n is the following holds. Polar cone to the cone K ⊂ R holds. Polar cone to the cone K ⊂ R is the following subset subset subset subset 0 .. n 0 = n | p, x ≤ 0, K ∀x (1) ∈ R .. {p 0 n | p, x ≤ 0, K ∀x ∈ ∈ K}. K}. (1) {p ∈ R 0 = K (1) = {p ∈ R K = {p ∈ Rn || p, p, x x ≤ ≤ 0, 0, ∀x ∀x ∈ ∈ K}. K}. (1) The normed linear space of absolutely continuous functions The normed linear space of absolutely continuous functions The normed linear space continuous functions nof The normedby linear space absolutely continuous functions nof we denote AC(T, R ). absolutely The derivative of absolutely n we denote by AC(T, R The derivative of absolutely n ). we denote by AC(T, R ). The derivative of absolutely n we denote by AC(T, R ). The derivative of absolutely n ) may continuous function x(·) ∈ AC(T, R not n ) may not exist continuous function x(·) ∈ AC(T, exist n continuous function x(·) ∈Therefore AC(T, R Rdetermine may not not exist continuous AC(T, R )) may on a set set of of function measure x(·) zero. ∈ at exist each on a measure zero. Therefore determine at each on a set of measure zero. Therefore determine at each on a set of measure zero. Therefore determine at each point t of interval T a generalized (maybe multi-valued) point t of interval T a generalized (maybe multi-valued) point t of interval T a generalized (maybe multi-valued) point t of interval T a generalized (maybe multi-valued) derivative D(x(·), t), which coincides up to a factor with derivative D(x(·), t), which coincides up to a factor with derivative D(x(·), which coincides up to with derivative D(x(·), t), t), which coincides up to aa factor factor with classical derivative at the points where classical derivative classical derivative at the points where classical derivative classical derivative at the points where classical derivative classical derivative at the points where classical derivative exists. The The set set D(x(·), D(x(·), t) t) is is determined determined by by the the formula formula exists. .. set exists. The D(x(·), t) is determined by the formula exists. The D(x(·), t) = .. set D(x(·), t) is determined by the formula D(x(·), t) = D(x(·), t) = D(x(·), t) = x(t) − x(t − λ) . (2) x(t) − x(t − λ) (0) + B x(t) − x(t − λ) ε x(t) − x(t − λ) (0) .. (2) + B ε x(t) − x(t − λ) + λ + B (0) (2) ε (0) . (2) + B x(t) − x(t − λ) + λ ε ε>0 δ∈(0,t−t0 ) λ∈(0,δ) x(t) − x(t − λ) + λ ε>0 δ∈(0,t−t0 ) λ∈(0,δ) x(t) − x(t − λ) + λ ε>0 δ∈(0,t−t0 ) λ∈(0,δ) ε>0 δ∈(0,t−t0 ) λ∈(0,δ)
n n ), which is R We also define aa normed space AC ∞ (T, n R which is We also define normed space AC ∞ (T, n ), (T, R ), which We also define a normed space AC n ∞ (T, R ),continuous which is is We also define a space AC nnormed subset of AC(T, R ), consisting of absolutely ∞ n subset of AC(T, R ), consisting of absolutely continuous n ), consisting subset of AC(T, AC(T, R→ ofderivative absolutelysatisfies continuous n subset of R ), consisting absolutely continuous n , whose of functions f : T R the functions → R ,, nwhose derivative satisfies the functions ffff:::(·)T T∈ L → (T, RnnR whose derivative satisfies the functions T → R , nnwhose derivative satisfies the condition ), and the norm is determined ∞ (T, condition ff (·) ∈ L ), and the norm is determined ∞ (T, R . n condition (·) ∈ L R ), and the norm is determined ∞ (T, = . condition f (·) ∈ L R ), and the norm is determined n by the formula f ) + f . f (t ∞ AC∞ = Rn + f L∞ . by the formula f AC .. f (t00 ) L∞ by f )R AC∞ Rn n + f L∞ . ∞ = f (t0 by the the formula formula f n n AC∞ = f (t0 )R + f L∞ . n ) (F(Rn )) we denote the set of all (non-empty By P(R n n By P(R we denote the set of all (non-empty n) n )) By P(R )) (F(R (F(R )) denote of (non-empty n the By P(Rsubsets (F(R )) we we denote the set set(see of all all (non-empty n . Recall closed) of the space R Polovinkin and n closed) subsets of the space R . Recall (see and n . Recall (see Polovinkin closed) subsets of the space R Polovinkin and closed) subsets of the space R . Recall (see Polovinkin and Balashov (2007) §1.4 and Polovinkin (2014) §24), that the that the Balashov (2007) §1.4 and Polovinkin (2014) §24), Balashov (2007) §1.4 and Polovinkin (2014) §24), that the Balashov (2007) §1.4 and Polovinkin (2014) §24), that the lower tangent cone(also called: the simple tangent cone) to lower tangent cone(also called: the simple tangent cone) to lower tangent cone(also called: the simple tangent cone) to n called: the simple tangent cone) to lower tangent cone(also n at the point a ∈ A is the following subset the set A ⊂ R n at the point a ∈ A is the following subset the set A ⊂ R n the at the then point point a a∈ ∈A A −1 is the the following following subset subset .. at the set set A A⊂ ⊂R R is (A; a) a) = = {v ∈ ∈R Rnn || lim lim (v, (v, λ λ−1 (A − − a)) a)) = = 0}. 0}. (3) (3) T .. {v L (A; −1 (A T λ↓0 (v, TL L (A; (A; a) a) = = {v {v ∈ ∈R Rn || lim lim λ−1 (A (A − − a)) a)) = = 0}. 0}. (3) (3) T λ↓0 (v, λ L λ↓0 λ↓0
n n n → P(Rn ). Consider aa multi-valued mapping F :: T × R n n Consider multi-valued mapping F T × R P(R n → n ). Consider a multi-valued mapping F : T × R → P(R ). Consider a multi-valued mapping F : T × RF → P(Rset ). Recall that effective set of the mapping is the Recall that effective set of the mapping F is the set . Recall that effective set of the mapping F is the set . Recall that effective set of the mapping F is the set domF = {(t, x) | F (t, x) = ∅}. Graph of the mapping . domF = . {(t, domF = {(t, x) x) ||| F F (t, (t, x) x) = = ∅}. ∅}. Graph Graph of of the the mapping mapping domF = {(t, x) F (t, x) = ∅}. Graph of the mapping is the set is the set is the set .. is the set n n n × Rn | y ∈ F (t, x)}. graph F = x, y) ∈ T × R .. {(t, n graph F = {(t, x, y) ∈ T × R Rnn || yy ∈ F (t, x)}. n× graph F = {(t, x, y) ∈ T × R × graph F = {(t, x, y) ∈ T × R ×R R |y∈ ∈F F (t, (t, x)}. x)}. Also we call Also we call Also we .. Also graph we call callF (t, ·) = n n graph F (t, ·) = {(x, y) y) ∈ ∈R Rnnn × ×R Rnnn ||| yyy ∈ ∈F F (t, (t, x)}. x)}. .. {(x, graph graph F F (t, (t, ·) ·) = = {(x, {(x, y) y) ∈ ∈R R × ×R R |y∈ ∈F F (t, (t, x)}. x)}. n n n → P(Rn ) the Suppose that for aa mapping F :: T × R n Suppose that for mapping F T × R → P(R ) the n → P(Rn Suppose that for a mapping F : T × R Suppose that for a mapping Fis : defined T × R → P(Rn )) the the following differential inclusion following differential inclusion is defined following differential differential inclusion inclusion is is defined defined following
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150 150 150 150
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dx ∈ F (t, x), dt
t ∈ T.
(4)
Absolutely continuous function x(·) ∈ AC(T, Rn ), which . n is integrable branch of derivative x = dx dt : T → R the mapping F (·, x(·)) is called trajectory of differential inclusion (4) on segment T . By RT (F, C0 ) we denote the subset from AC(T, Rn ) of all the trajectories x(·) of differential inclusion (4) on the segment T provided that x(t0 ) ∈ C0 .
By R∞ T (F, C0 ) we denote the subset of RT (F, C0 ) consisting of all the trajectories x(·) from the space AC∞ (T, Rn ). 3. MEASURABLE-PSEUDO-LIPSCHITZ CONDITIONS
Hypothesis 1. Suppose that the mapping F : T × Rn → P(Rn ) such that there exist such a number δ > 0, . functions l(·) ∈ L∞ (T, R1+ ) and η(t) = δ(1 + 2l(t) exp((t − t0 )l(·)L∞ )), t ∈ T , and the closed subset W , satisfying inclusion: W ⊃ {(t, x) ∈ T × Rn | x − x (t) ≤ t ≤ 4δ exp((t − t0 )l(·)L∞ ) + 3 η(τ )dτ }, t0
that the following three conditions hold: ∀(t, x) ∈ W (5)
are non-empty, 2) for any function v(·) ∈ C(T, Rn ), for which graph v ⊂ W , the mapping t → G(t, v(t)) is measurable,
3) for any points (t, x1 ), (t, x2 ) ∈ W inclusions hold
Further we will need a result on the differentiation of the set of trajectories of differential inclusion with initial data from the paper Polovinkin (2013). Let the mapping F : T × Rn → F (Rn ) and the trajectory x (·) ∈ RT (F, C0 ) be defined. Define for any t ∈ T the following mapping . Ft : Rn → F(Rn ), where Ft (x) = F (t, x), and consider its lower multi-valued derivative at the point ( x(t), x (t)) ∈ n n n R × R in the direction u ∈ R (see Polovinkin (2013)) of the following type . . FL (t, u) = lim inf λ−1 (F (t, x (t) + λu) − x (t)) = λ↓0 . [λ−1 (F (t, x (t) + λu) − x (t)) + Bε (0)]. =
Similarly with notation RT (F, x0 ) for the set of trajectories of inclusion (4) by symbol RT (FL , u0 ) we denote the set of all solutions of Cauchy problems of the corresponding differential inclusion u (t) ∈ FL (t, u(t)),
t ∈ T,
(8)
with the initial condition u(t0 ) = u0 . From Polovinkin (2013) (see Theorems 3 and 4 and Corollary 5) and condition l(·) ∈ L∞ (T, R1+ ) one can obtain the following result. Theorem 1. Let subset C0 ⊂ Rn , mapping F : T × Rn → F(Rn ) and trajectory x (·) ∈ RT (F, C0 ) be given. Suppose that Hypothesis 1 is true at the neighborhood of trajectory x (·). Then for any u(·) ∈ RT (FL , TL (C0 , x (t0 ))), for which u (·) ∈ L∞ (T, Rn ), there exists such a number α > 0 that for any λ ∈ (0, α) there exist solution xλ (·) ∈ RT (F, C0 ) and a function o(λ, ·) from AC∞ (T, Rn ) such that (t) + λu(t) + o(λ, t), t ∈ T, xλ (t) = x −1 where lim λ o(λ, ·)AC∞ = 0. λ→+0
G(t, x1 ) ⊂ F (t, x2 ) + l(t)x1 − x2 B1 (0).
Definition 1. In the case when a mapping F : T × Rn → P(Rn ) satisfies Hypothesis 1, we say that the mapping F is called measurably-pseudo-Lipschitz in the neighborhood of trajectory x (·).
Hypothesis 2. Let the mapping F : T × Rn → P(Rn ) be measurable-pseudo-Lipschitz in the neighborhood of trajectory x (·) ∈ RT (F, x 0 ). Suppose that for subset of the following type . x) = G(t, F (t, x) ∩ ( x (t) + 3η(t)B1 (0)), ∀(t, x) ∈ W, (6) the next conditions hold:
2’) for any function v(·) ∈ C(T, R ) with graph v ⊂ W , v(t)) is measurable, the mapping t → G(t, n
3’) for any (t, x1 ), (t, x2 ) ∈ W inclusions hold x1 ) ⊂ F (t, x2 ) + l(t)x1 − x2 B1 (0). G(t,
4. DIFFERENTIATION OF THE SET OF TRAJECTORIES
ε>0 δ>0 λ∈(0,δ)
0 ) of differential incluFix some trajectory x (·) ∈ RT (F, x sion (4).
1) the subsets . G(t, x) = F (t, x) ∩ ( x (t) + η(t)B1 (0)),
151
(7)
Definition 2. In the case when a mapping F : T × Rn → P(Rn ) satisfies Hypothesis 2, we say that the mapping F is called strictly measurably-pseudo-Lipschitz in the neighborhood of trajectory x (·).
151
5. CONTINUOUS DEPENDENCE OF THE TRAJECTORIES ON THE INITIAL APPROXIMATIONS In the book Polovinkin (2014) there are some results on the continuous dependence of trajectories of the differential inclusion on initial approximations. Here are two results from §44 of this work. Let the mapping F : T × Rn → P(Rn ) satisfies Hypotheses 1 – 2 in the neighborhood of trajectory x (·) ∈ RT (F, x 0 ).
It means that the number δ ∈ (0, 1), functions l(·) and and the closed set W ⊂ T × Rn η(·), the mappings G, G are given.
Define a function m : T → R1+ of the following type . t m(t) = t0 l(τ ) dτ .
a) Consider an arbitrary function ρ0 (·) ∈ L1 (T, R1+ ) with ρ0 (t) ≤ η(t) for almost all t ∈ T . In the space AC(T, Rn ) define a subset . A(F, ρ0 (·)) = z(·) ∈ AC(T, Rn )z(t0 ) − x 0 ≤ δ,
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(z (t), F (t, z(t))) ≤ ρ0 (t), z (t) − x (t) ≤ η(t)
t∈T .
It is easy to show that the subset A(F, ρ0 (·)) is nonempty and graphs of all curves of this subset are contained in W . Moreover, for any z0 ∈ Bδ ( x0 ) there exists such z(·) ∈ A(F, ρ0 (·)) that z(t0 ) = z0 . . Theorem 2. Define a subset S0 = A(F, ρ0 (·)), where the function ρ0 (·) is determined above. Let the function d : Bδ ( x0 ) → Rn be continuous and such that d(z) − z < δ for all z ∈ Bδ ( x0 ). Define a finite set of trajectories . X = {wi (·)}Ii=1 ⊂ RT (G, Bδ ( x0 )) of the differential inclusion which is contained in the set S0 and d(wi (t0 )) = wi (t0 ) for all i ∈ 1, I. Then for any ε > 0 there exists a continuous mapping r: S0 → d(Bδ ( RT (G, x0 )), which satisfies: r(z(·))(t0 ) = d(z(t0 )) ∀z(·) ∈ S0 , r(wi (·))(·) = wi (·)) ∀i ∈ 1, I, t z (s) − d r(z(·))(s) ds ≤ ξε (t) ∀ t ∈ T, z(·) ∈ S0 , ds
t0
t
. b) We denote the standard simplex in R by Σ = k+1 . . γm = 1 . (9) = γ = (γ1 , . . . , γk+1 ) ∈ Rk+1 | γm ≥ 0, k
m=1
Theorem 3. Suppose that a subset C0 ⊂ Rn and a mapping F : T × Rn → F(Rn ), satisfying Hypothesis 2 near some trajectory x (·) ∈ RT (F, C0 ) are given. Let T0 be arbitrary regular tangent to the subset C0 at the point x (t0 ) cone. Suppose that finite set of trajectories um (·) ∈ R∞ T (FL , T0 ), m ∈ 1, k + 1 is given. Define k+1 . uγ (·) = γm um (·), γ ∈ Σk . m=1
(10)
Then there exists such a number α > 0 that for any λ ∈ (0, α) and for any γ ∈ Σk there exist a trajectory xλ,γ (·) ∈ RT (F, C0 ) and a function t → o(λ, γ, t) from AC∞ (T, Rn ) such that (t) + λuγ (t) + o(λ, γ, t), t ∈ T, xλ,γ (t) = x −1 lim max λ o(λ, γ, ·) AC∞ = 0, λ→+0
γ∈Σk
t0
t1 t
x∗ (s) ds ∈ K 0 (t),
a.e. t ∈ T. (14)
Consider the following extremal problem on the segment . T = [t0 , t1 ]:
t0
Let the following inclusion holds uγ (·) | γ ∈ Σk ⊂ RT (FL , T0 ).
7. TIME OPTIMUM PROBLEM
e−m(τ ) ρ0 (τ )dτ (1 + ε)) + ε. k+1
a.e.t ∈ T. Q(t, x) ∩ (γ(t)B1 (0)) = ∅, ∀x ∈ B1 (0), 0 0 Let K0 and K (t) be polar cones to the cones K0 and 0 K(t) respectively. Then the polar cone (R∞ T (F, K0 )) to ∞ ∗ ∗ the set RT (F, K0 ) consists of pairs of points b ∈ E and functions y ∗ (·) ∈ L1 (T, E ∗ ) such that for each pair there is a function x∗ (·) ∈ L1 (T, E ∗ ), for which the following inclusions hold t1 ∗ (13) b − x∗ (s) ds ∈ K00 ; x∗ (t), y ∗ (t) −
where
. ξε (t) = em(t)+ε (δ +
Theorem 4. Let K0 be a closed convex cone in a separable reflexive Banach space E. Let Q: T × E → F(E) be . such that Q(t, x) = {y ∈ E | (x, y) ∈ K(t)}, where the set K(t) is a closed convex cone in space E × E and it’s measurably time-dependent. Suppose that there is a function γ(·) ∈ L∞ (T, R1+ ) such that
(11)
(12)
besides, mappings γ → o(λ, γ, ·) from Σk to AC∞ (T, Rn ) are continuous. 6. THE POLAR OF A CONVEX PROCESS Due to Polovinkin (2012) one can calculate the polar cone to cone of trajectories of differential inclusion, the right part of which has a convex conical graph. 152
Minimize {t ∈ (t0 , t1 ] | x(·) ∈ RT (F, C0 ), x(t) ∈ C1 }. (15) where C0 , C1 ⊂ Rn are disjoint closed subsets.
In other words among all trajectories RT (F, C0 ) of differential inclusion (4), we are to find such a trajectory for which time τ of hitting the final subset C1 is minimal. . Suppose that such a time t = min τ and a trajectory x (·) exist. Then value t will be called optimal time, and x (·) solution of time optimum problem. Let the trajectory x (·) ∈ RT (F, C0 ) be solution of time optimum problem (15) and t - optimal time.
Formulate conditions on the mapping F in the problem (15). Suppose that Hypotheses 1 – 2 are satisfied. Hypothesis 3. Measurable mapping K : T → F(Rn × Rn ) is given, which values are closed convex cones satisfying inclusion K(t) ⊂ TL (graph F (t, ·); ( x(t), x (t))) a.e. t ∈ T, where TL (A, a) — the lower tangent cone to the set A at the point a ∈ A (see (3) ). Examples of mapping K(t) satisfying Hypothesis 3 are:
1) Clarke’s tangent cone to the set graph F (t, ·) at the point ( x(t), x (t)) (see §2.4 Clarke (1983));
2) asymptotic lower tangent cone to the set graph F (t, ·) at the point ( x(t), x (t)) (see §1.4 Polovinkin and Balashov (2007)).
Method of study of time optimum problem consists in the fact that using the optimal solution of the problem we investigate the set of trajectories of the differential inclusion in variations (8), which by virtue of Theorems 1 – 3 forms tangent cone to the set of trajectories of the original inclusion. Due to Theorem 4 we calculate the
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polar cone to cone of trajectories of differential inclusion in variations. Due to the optimality of a trajectory x (·) we show that this polar cone is in some sense a normal cone to a subset of trajectories of the original differential inclusion, from which we obtain necessary conditions optimality in the Lagrangian form. Theorem 5. Let the mapping F satisfy Hypotheses 1 – 3 at the neighborhood of x (·), also x (·) is the local solution of the time optimum problem (15) in the space AC(T, Rn ) and t is optimal time. Let K(t) from Hypothesis 3. Let K0 and K1 be some regular tangent cones to the sets C0 and C1 at the points x (t0 ) and x ( t) respectively. Then for any ξ ∈ D( x(·), t) (see (2)) there exists such a function t], Rn ), p(·) = 0 that the following formulae p(·) ∈ AC([t0 , hold: t) ∈ −K10 , (16) p(t0 ) ∈ K00 , p( p( t), ξ ≥ 0,
(p (t), p(t)) ∈ K 0 (t)
a.e. t ∈ [t0 , t],
(17)
(18)
where K00 , K10 , K 0 (t) – polar cones to the cones K0 , K1 , K(t) respectively (see (1)). 8. PROOF OF THE MAIN RESULT To prove Theorem 5 we transform original boundary value problem (15), defined in space Rn , to Cauchy problem in extended space Rn × Rn , in which for a point . y = (y 1 , y 2 ) ∈ Rn × Rn . we determine the norm y = max(y 1 , y 2 ). Define a . set C˜ = (C0 , C1 ) ⊂ Rn × Rn , a function . y (t) = ( x(t), x ( t)) (19)
: T × (Rn × Rn ) → P(Rn × Rn ) by and a mapping G formula . y) = G(t, {(v 1 , 0) ∈ Rn × Rn | v 1 ∈ G(t, y 1 )}. Define also a linear operator Λ : Rn × Rn → Rn of the following type . Λy = y 1 − y 2 . Due to optimality of the pair ( x(·), t) we have 0 = Λ y (t), 0 ∈ / Λ(A|t ) for all t ∈ [t0 , t), (20) where
. C)}. ˜ A|t = {y(t) ∈ (Rn × Rn ) | y(·) ∈ RT (G,
(21)
Determine mappings Q, P : T ×(Rn ×Rn )×R1 → P((Rn × Rn ) × R1 ) of the following type . Q(t, y, y 0 ) = y))}, (22) {(v, v 0 ) ∈ (Rn × Rn ) × R1 | v 0 ≥ (v, G(t, . P (t, y, y 0 ) = {(v, v 0 ) ∈ (Rn × Rn ) × R1 | v 0 ≥ (1 + l(·)L∞ ) ((y, v), R(t))}, . where R(t) =
{(y 1 , y 2 , v 1 , 0) ∈ Rn × Rn × Rn × Rn | (y 1 , v 1 ) ∈ K(t)}, and l(·) is from Hypothesis 1. It is easy to check that mappings Q and P are measurable in t ∈ T and for every t ∈ T are Lipschitz on y, y 0 in some neighborhood of the 153
153
point ( y (t), 0), the sets R(t), graph P (t, ·) are convex and closed. Denote . C = C˜ × {0} ⊂ (Rn × Rn ) × R1 , . D0 = K0 × K1 × {0} ⊂ Rn × Rn × R1 . In the same way as we did in §4, for almost every t ∈ T we introduce the notation of lower derivative of the mapping y (t), 0, y (t), 0) ∈ (y, y 0 ) → Q(t, y, y 0 ) at the point ( 0 graph Q(t, ·) in the direction (u, u ) ∈ (Rn × Rn ) × R1 of the following type . Q L (t, u, u0 ) = {(v, v 0 ) ∈ (Rn × Rn ) × R1 | (u, u0 , v, v 0 ) ∈ y (t), 0, y (t), 0))}. ∈ TL (graphQ(t, ·); ( t] and for any It is easy to show that for a.e. t ∈ [t0 , (u, u0 ) ∈ (Rn × Rn ) × R1 the following inclusion holds P (t, u, u0 ) ⊂ Q L (t, u, u0 ).
(23)
For an arbitrary vector ξ ∈ D( x(·), t) (see (2)) we define a set MΛ in the space Rn × R1 as follows . MΛ = {(τ ξ + Λu( t)) ∈ Rn × R1+ | τ ≤ 0, t), u0 ( (u(·), u0 (·)) ∈ RT (P, D0 ),
(u (·), u0 (·)) ∈ L∞ (T, (Rn × Rn ) × R1 )}.
(24)
Obviously that is convex cone. Prove that polar cone MΛ0 contains such point (q, q 0 ) ∈ Rn × R1 that q = 0. Assume the contrary. i.e. let MΛ0 ⊂ {(0, α) ∈ Rn × R1 | α ∈ R1 }.
(25)
Then MΛ00 = M Λ ⊃ {(x, 0) | x ∈ Rn }, whence by the convexity of the set MΛ we have the following inclusion (26) MΛ ⊃ Rn × (R1+ \ {0}). Based on the standard simplex Σn (see (9)), chose in Rn a simplex σ of the following type n+1 . . σ = {z = z(γ) | z(γ) = γi zi , γ ∈ Σn }, i=1
where zi ∈ Rn , i ∈ 1, n + 1 are the vertices of this simplex, such that 0 ∈ int σ. Let Γ be boundary of the simplex σ and . a = min{z | z ∈ Γ}. It follows from the inclusion 0 ∈ int σ that a > 0. Consider convex set . L = Λu( t) | (u(·), u0 (·)) ∈ RT (P, D0 ), (u (·), u0 (·)) ∈ a t) ≤ e−m( t)−1 . ∈ L∞ (T, (Rn × Rn ) × R1 ), u0 ( 8
From the definitions of the subset L, the inclusion (26) and the definition of subset MΛ (24) follows the equality (τ ξ + L) = Rn . (27) τ ≤0
From this equality and convexity of the set L follows monotonicity in τ ≤ 0 of the subsets τ ξ + L, i.e. for any τ1 , τ2 ∈ R1 , τ1 ≤ τ2 ≤ 0, the inclusion holds τ2 ξ + L ⊂ τ1 ξ + L. If ξ = 0 then the statement is obvious. If ξ = 0, then assume the contrary to prove the statement . Suppose that there exist such τ1 , τ2 ∈ R1 , τ1 < τ2 ≤ 0 that
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τ2 ξ+L τ1 ξ+L. Then there exists such point m0 ∈ L that . m0 ∈ / τ3 ξ + L, where τ3 = τ1 − τ2 . By the theorem on the separability of convex sets there exists p ∈ Rn , p = 0, such that p, m0 ≥ τ3 p, ξ + s(p, L). Since p, m0 ≤ s(p, L), then p, ξ ≥ 0 and s(p, L) < +∞. Therefore for any τ ≤ 0 we get that s(p, τ ξ+L) = τ p, ξ+s(p, L) ≤ s(p, L) < +∞, i.e. the following inclusion holds . (τ ξ + L) ⊂ H = {x ∈ Rn | p, x ≤ s(p, L) < +∞}, τ ≤0
and besides H = Rn , which contradicts the equality (27).
From monotonicity in τ ≤ 0 of the sets τ ξ + L, the equality (27) and the fact that the simplex σ is bounded it follows that there exists such τ0 ≥ 1 that the following inclusion holds σ ⊂ −τ0 ξ + L. Therefore there exist trajectories (vi (·), vi0 (·)) ∈ RT (P, D0 ), i ∈ 1, n + 1 with significantly bounded derivatives such that a t) ≤ e−m( t)−1 , −τ0 ξ + Λvi ( t) = zi , (28) vi0 ( 8 t where m(t) = t0 l(s) ds. For any γ ∈ Σn (see (9)) we define in AC(T, (Rn × Rn ) × R1 ) functions n+1 . γi · (vi (·), vi0 (·)), (vγ (·), vγ0 (·)) = i=1
γ ∈ Σn .
(29)
Due to the convexity of the subset RT (P, D0 ) we obtain inclusions (vγ (·), vγ0 (·)) ∈ RT (P, D0 ), ∀ γ ∈ Σn .
Hence and from the inclusion (23) by Theorem 3 there exists such a number δ0 > 0 that for all α ∈ (0, δ0 ) and 0 for all γ ∈ Σn there exist trajectories (yα,γ (·), yα,γ (·)) ∈ RT (Q, C) such that . yα,γ (t) = y (t) + αvγ (t) + o(t, α, γ), . 0 yα,γ (t) = αvγ0 (t) + o0 (t, α, γ), moreover lim max α−1 (o(·, α, γ), o0 (·, α, γ))AC = 0,
α→0 γ∈Σn
(30)
0 and functions γ → (yα,γ (·), yα,γ (·)) from Σn to AC(T, (Rn n 1 ×R )×R ) are continuous and uniformly bounded by some additive function for all α, γ. Therefore due to inequalities that follow from the definition (22) of mapping Q 0 yα,γ (t))) yα,γ (t) ≥ (y α,γ (t), G(t,
(31)
by Theorem 2 there exist δ1 ∈ (0, δ0 ) and trajectories yα,γ (t0 )), α ∈ (0, δ1 ) zα,γ (·) ∈ RT (G,
which continuously depend on γ ∈ Σn , moreover due to estimates from Theorem 2 the next inequalities are true t αa αa m( t)+1 0 yα,γ (s) ds + ≤ , zα,γ (t) − yα,γ (t) ≤ e 8 4 t0
t ∈ [t0 , t], α ∈ (0, δ1 ). In turn from the equality (30) there exists δ2 ∈ (0, δ1 ] such that αa a.e. t ∈ [t0 , t], ∀ α ∈ (0, δ2 ), γ ∈ Σn . o(t, α, γ) < 4
154
Thus for all α ∈ (0, δ2 ) and for all γ ∈ Σn the function zα,γ (·) can be represented as follows zα,γ (t) = y (t) + αvγ (t) + αδ(t, α, γ), (32)
where functions γ → δ(·, α, γ) are continuous on Σn , and inequalities hold true a δ(·, α, γ)C < , ∀α ∈ (0, δ2 ), γ ∈ Σn . 2 Due to the definitions of the vector ξ ∈ D( x(·), t) (see (9)) and the function y (·) (see (19)) there exist . sequences λk → 0, β k = (βk , 0) ∈ Rn × Rn , βk → 0 such that y ( t) = y ( t − λk ) + αk ζ + αk β k , where . ζ = (ξ, 0) ∈ Rn × Rn and . αk = t) + λk . y ( t − λk ) − y ( Rewrite the last equality, substituting in it y ( t − λk ) from equality (32), where t = t − λk and α = αk /τ0 , τ0 is obtained in (28). Thus we have αk αk y ( t) + (−τ0 ζ + vγ ( t)) = z αk ,γ ( t − λk ) + ηk (γ), (33) τ 0 τ0 τ0 where the functions αk . ηk (γ) = τ0 βk + vγ (t) − vγ (t − λk ) − δ t − λk , ,γ τ0 are continuous on Σn and there exists such a number k0 that maxn ηk0 (γ) < a/2. γ∈Σ
Since γ = (γ1 , . . . , γn+1 ) ∈ Σn are barycentric coordinates on the simplex σ, then the function η: σ → E × E of . the following type η(z) = ηk0 (γ) (provided z = z(γ)) is continuous on σ. Consider a continuous function f : σ → Rn of the type: . f (z) = z − Λη(z). By the construction for any z ∈ Γ we have z ≥ a, i.e. for all z ∈ Γ the following inequality holds f (z) − z ≤ 2 max{ηk0 (γ) | γ ∈ Σn } < a ≤ z − 0. Hence by the theorem (scholium) in Sec. 4.1 in the book Lee and Markus (1972) the inclusion 0 ∈ f (σ) holds, that is there exists such a point z0 = z(γ 0 ) ∈ σ that f (z0 ) = 0. By virtue of equalities (28) and (29) provided γ 0 ∈ Σn we have an expression z(γ 0 ) = −τ0 ξ + Λvγ 0 ( t), from which the equality f (z0 ) = 0 takes form (34) t) = Ληk0 (γ 0 ). −τ0 ξ + Λvγ 0 ( Applying the operator Λ to both sides of equality (33) and reducing two terms due to equality (34), we obtain the following equality Λ y ( t) = Λz αk0 ,γ 0 ( t − λk0 ). τ0
Since Λ y ( t) = 0 (see (20)), then t − λk0 ) = 0, Λz αk0 0 ( τ0
,γ
that is there exists a trajectory of differential inclusion which hits the final set at the moment t − λk0 , which
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contradicts the optimality of the pair ( x(·), t) (see (20), (21)).
Thus it is proved that assumption (25) was wrong and there exists a point (q, q 0 ) ∈ MΛ0 with q = 0, that is due to (24) for any τ ≤ 0 and for any (v(·), v 0 (·)) ∈ RT (P, D0 ) the inequality holds t) + q 0 v 0 ( t) ≤ 0. Λ∗ q, τ ζ + v(
(35)
Note that the cone graphP (t, ·) is convex and closed, measurably depends on t on [t0 , t1 ], and satisfies the conditions of Theorem 4. Choosing τ = 0 in (35) we obtain that the function η ∗ = (b∗ , y ∗ (·)), where b∗ = y ∗ (t) ≡ (Λ∗ q, q 0 ) ∈ (Rn × Rn ) × R1 , on the segment [t0 , t] belongs to the cone (R[t0 , t] (P, D0 ))0 , whence by Theorem 4 for the function η ∗ = (b∗ , y ∗ (·)) t], (Rn × Rn ) × R1 ), there exists function x∗ (·) ∈ L1 ([t0 , for which inclusions (13), (14) hold. Choosing a function (¯ p(·), p¯0 (·)) ∈ AC(T, (Rn × Rn ) × R1 ) of the type . (¯ p(t), p¯ (t)) = b∗ − 0
t
x∗ (s)ds,
t
from inclusions (14), (13) we obtain (¯ p (t), p¯0 (t), p¯(t), p¯0 (t)) ∈ {τ (u∗ , 0, v ∗ , −1) | τ ≥ 0,
t]; (u∗ , v ∗ ) ∈ (1 + l(·)∞ )(R0 (t) ∩ B1R (0))}, t ∈ [t0 , p( t), p¯0 ( t)) = (Λ∗ q, q 0 ). (¯ p(t0 ), p¯0 (t0 )) ∈ D00 , (¯ 4n
In (35) choosing a trajectory (v(·), v 0 (·)) = 0, we obtain Λ∗ q, ζ ≥ 0, that is ¯ p( t), ζ ≥ 0.
Finally, representing p¯(t) ∈ Rn × Rn in the form p¯(·) = (p1 (·), p2 (·)), where p1 (t), p2 (t) ∈ Rn , we get p1 (t0 ) ∈ K00 ,
p1 ( t), ξ ≥ 0;
p2 (t0 ) ∈ K10 ,
p2 ( t) = −p1 ( t) = q, 1 1 0 (p (t), p (t)) ∈ K (t), t ∈ [t0 , t]. . Thus supposing that p(·) = p1 (·), we obtain that p(·) = 0 (since p( t) = −q = 0) and the formulae (16)–(18) hold true. p2 (t) ≡ 0,
REFERENCES J.P. Aubin, Lipschitz behavior of solutions to convex minimization problems, Math. of Oper. Res. 9, pp. 87 – 111 (1984). Ph. D. Loewen and R. T. Rockafellar. Optimal control of unbounded differential inclusions, SIAM J. Control and Optimization. 32 (2), pp. 442 – 470 (1994). A.D. Ioffe, Existence and relaxation theorems for unbounded differential inclusions, J. Convex Anal. 13 (2), pp. 353 – 362 (2006). F.H. Clarke, Optimization and Nonsmoth Analysis, (J. Wiley, New York, 1983) E. S. Polovinkin and G. V. Smirnov, An approach to the differentiation of many-valued mappings, and necessary conditions for optimization of solutions of differential inclusions, Differential Equations 22 (1986), pp. 660 – 668. Russian original in Differen. Uravn. 22 (1986), pp. 944 - 954. E. S. Polovinkin and G. V. Smirnov, Time-optimum problem for differential inclusions, Differential equations 22 (1986), pp. 940 – 952. Russian original in Differen. Uravn. 22 (1986), pp. 1351 – 1365.
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E. S. Polovinkin, The properties of continuity and differentiation of solution sets of Lipschitzean differential inclusions, in Modeling, Estimation and Control of Systems with Uncertainty, Ed. by G. B. Di Masi, A. Gombani, and A. B. Kurzhansky (Birkh¨ auser, Boston, 1991), Prog. Syst. Control Theory 10, pp. 349 - 360. E. S. Polovinkin, Necessary conditions for optimization problems with differential inclusions, in Set-Valued Analysis and Differential Inclusions, Ed. by A. B. Kurzhanski and V. M. Veliov (Birkh¨ auser, Boston, 1993), Prog. Syst. Control Theory 16, pp. 157 - 170. E. S. Polovinkin, Necessary conditions for an optimization problem with a differential inclusion, Proc. Steklov Inst. Math. 211, pp. 350 - 361 (1995). E. S. Polovinkin and M. V. Balashov, Elements of Convex and Strongly Convex Analysis, 2nd ed. (Fizmatlit, Moscow, 2007) [in Russian]. E. S. Polovinkin, On the calculation of the polar cone of the solution set of a differential inclusion, Proc. Steklov Inst. Math. 278, pp. 169 - 178 (2012). E. S. Polovinkin, Differential Inclusions with Measurable PseudoLipschitz Right-Hand Side, Proc. Steklov Inst. Math. 283, pp. 116 - 135 (2013) E. S. Polovinkin, Set-Valued Analysis and Differential Inclusions, (Fizmatlit, Moscow, 2014) [in Russian]. E. B. Lee and L. Markus, Foundations of Optimal Control Theory, (Wiley, New York, 1967).
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Time Optimum Problems for Time Unbounded for Unbounded Time Optimum Optimum Problems Problems for Unbounded Differential Inclusion Differential Inclusion Differential Inclusion ∗ ∗ ∗ ∗
Moscow Moscow Moscow Moscow
∗ Evgenii S. Evgenii S. Polovinkin Polovinkin ∗∗∗ Evgenii Evgenii S. S. Polovinkin Polovinkin Institute of Physics and Technology, Moscow, Institute of Physics and Technology, Moscow, Institute of Physics and Technology, Moscow, Institute of Physics and Technology, Moscow, (e-mail: [email protected]). (e-mail: [email protected]). (e-mail: [email protected]). [email protected]). (e-mail:
Russia, Russia, Russia, Russia,
Abstract: Based on the properties of solutions of a differential inclusion with unbounded Abstract: of solutions of inclusion with Abstract: Based Based on on the the properties properties of side, solutions of aaa differential differential inclusion with unbounded unbounded Abstract: Based on the properties of solutions of differential inclusion with unbounded measurable-pseudo-Lipschitz right-hand the necessary conditions in time optimum problem measurable-pseudo-Lipschitz right-hand side, the necessary conditions in time optimum problem measurable-pseudo-Lipschitz right-hand side, the necessary conditions in time optimum problem measurable-pseudo-Lipschitz right-hand side, the necessary conditions in time optimum problem on set of solutions of the unbounded differential inclusion are obtained. on set of solutions of the unbounded differential inclusion are obtained. on set of solutions of the unbounded differential inclusion are obtained. on set of solutions of the unbounded differential inclusion are obtained. DOI: 10.1134/S0081543813080099 DOI: 10.1134/S0081543813080099 DOI: DOI: 10.1134/S0081543813080099 10.1134/S0081543813080099 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: differential differential inclusion, inclusion, necessary necessary optimality optimality conditions, conditions, time time optimum optimum problem. problem. Keywords: Keywords: differential differential inclusion, inclusion, necessary necessary optimality optimality conditions, conditions, time time optimum optimum problem. problem. 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION 1. 1. INTRODUCTION In the paper Aubin (1984) there was conIn the the paper paper Aubin Aubin (1984) (1984) there there was was introduced introduced aaa conconIn In the of paper Aubin (1984) multi-valued there was introduced introduced awhich condition pseudo-Lipschitz mapping, dition of pseudo-Lipschitz multi-valued mapping, which dition of pseudo-Lipschitz multi-valued mapping, which dition of pseudo-Lipschitz multi-valued mapping, which proved to be more convenient for research of unbounded proved to to be be more more convenient convenient for for research research of of unbounded unbounded proved proved to be mappings more convenient forLipschitz researchcondition. of unbounded multi-valued than the multi-valued mappings than the Lipschitz condition. multi-valued multi-valued mappings mappings than than the the Lipschitz Lipschitz condition. condition. Generalizations of J. -P. Aubin condition to the nonGeneralizations of J. -P. Aubin condition to the the nonnonGeneralizations of J. -P. Aubin condition to Generalizations of J. -P. Aubin condition to the nonautonomous case, i.e. case when the mapping F takes the autonomous case, i.e. case when the mapping F takes the autonomous case, case when mapping F the n i.e. autonomous case, i.e.P(R casennn ), when the mapping F takes takes and the n → P(R ), werethe first done Loewen Loewen and form F :: T × R n → were first done form F T × R n → P(Rn ), were first done Loewen and form F : T × R → P(R ), were first done Loewen and form F : T × R Rockafellar (1994) and later Ioffe (2006). In these works Rockafellar (1994) and later Ioffe (2006). In these works Rockafellar (1994) and later (2006). works Rockafellar (1994) and global later Ioffe Ioffe (2006). In In these these works some conditions of the pseudo-Lipschitz are given. some conditions of the global pseudo-Lipschitz are given. some conditions conditions of of the the global global pseudo-Lipschitz pseudo-Lipschitz are are given. given. some In this paper, we consider the differential inclusion with In this paper, we consider the differential inclusion with In this paper, we consider the differential inclusion with In this paper, we consider the differential inclusion with the other local conditions called conditions of measurablethe other local conditions called conditions of measurablethe other local conditions called conditions of measurablethe other local conditions called conditions of measurablepseudo-Lipschitz mappings, which first were proposed pseudo-Lipschitz mappings, which first were proposed pseudo-Lipschitz mappings, which first proposed pseudo-Lipschitz mappings, which first were were(2013)). proposed by the author (for example, see Polovinkin In by the author (for example, see Polovinkin (2013)). In by the author (for example, see Polovinkin (2013)). In by the author (for example, see Polovinkin (2013)). In these works, under the conditions of measurable-pseudothese works, under the conditions of measurable-pseudothese works, under the conditions of measurable-pseudothese works, under the conditions of measurable-pseudoLipschitz right-hand side of the differential inclusion many Lipschitz right-hand side of differential inclusion many Lipschitz right-hand side of the theexistence differential inclusion many Lipschitz right-hand side of the differential inclusion many results were received, namely: theorem for soluresults were received, namely: existence theorem for soluresults were received, namely: existence theorem for soluresults were received, namely: existence theorem for solutions, theorem on the relaxation of differential inclusion, tions, theorem on the relaxation of differential inclusion, tions, theorem on the relaxation of differential inclusion, tions, theorem on the relaxation of differential inclusion, theorem on the differentiation of the initial data and other theorem on the differentiation of the initial data and other theorem on the differentiation of data other theorem onof the differentiation of the the initial initial data and and other properties trajectorys of differential inclusion with unproperties of trajectorys of differential inclusion with unproperties of trajectorys of differential inclusion with unproperties of trajectorys of differential inclusion with unbounded right-hand side. bounded right-hand side. bounded right-hand side. bounded right-hand side. This work is devoted to the development of the results This work is to of results This work is devoted devoted to the the development development of the the results This work is devoted to the development of the results on the study of optimization problems with differential on the study of optimization problems with differential on the study of optimization problems with differential on the study of optimization problems with differential inclusions, obtained Polovinkin and Smirnov (1986a,b), inclusions, obtained Polovinkin and Smirnov (1986a,b), inclusions, obtained Polovinkin and Smirnov (1986a,b), inclusions, obtained (1991, Polovinkin and Smirnov (1986a,b), and later Polovinkin 1993, 1995, 2013, 2014), from and later Polovinkin (1991, 1993, 1995, 2013, 2014), from and later Polovinkin (1991, 1993, 1995, 2013, 2014), from and later Polovinkin (1991, 1993, 1995, 2013, 2014), from the case of the bounded right-hand side of the differential the case of the bounded right-hand side of the differential the case of the bounded right-hand side of the differential the case of the bounded right-hand side of the differential inclusion to to the the case case where where the the right right side side is is unbounded. unbounded. inclusion inclusion inclusion to to the the case case where where the the right right side side is is unbounded. unbounded. 2. MAIN MAIN NOTATIONS NOTATIONS AND AND DEFINITIONS DEFINITIONS 2. 2. 2. MAIN MAIN NOTATIONS NOTATIONS AND AND DEFINITIONS DEFINITIONS .. We denote line segment by T = [t Euclidean 0, t We denote line segment T = tt11 ]]] and and Euclidean .. n[t ..by We denote segment by T = [t00|,,, x and Euclidean nline 1 We denote segment by T = t ] and Euclidean nline n[t0 space by R . Let B (a) = {x ∈ R − a < r} and 1 . r n n space by R . Let B (a) = {x ∈ R | x − a r} and . r n . Let nBr (a) = {x ∈ Rn | x − a < space by R < r} and . space by R . Let B (a) = {x ∈ R | x − a < r} and . r n B (a) = {x ∈ R | x − a ≤ r}. The distance from . r (a) = n B {x ∈ R | x − a ≤ r}. The distance from . r (a) = {x ∈ Rnn | x − a ≤ r}. The B distance from n r B (a) = {x ∈ R | x − a ≤ r}. The from n to n distance the point x ∈ R the subset A ⊂ R we denote by r n the point subset A R denote by .. x n to the point x ∈ ∈ R R− to |the the subset A ⊂ ⊂ Rnn we wefunction denote for by the point x ∈ R to the subset A ⊂ R we denote by (x, A) = inf{x y y ∈ A}. The support . (x, A) = inf{x − y | y ∈ A}. The support function for . . (x, A) A) A =⊂ inf{x − y y | yydenote ∈ A}. A}. by Thes(p, support function for| n − .. function (x, = inf{x | ∈ The support for n we subset R will A) = sup{p, x n subset A ⊂ R will denote by s(p, A) = || . sup{p, x n we subset A ⊂ R we will by s(p, A) = x subset Ap ⊂ R will denote denote by s(p, A) R =nnn sup{p, sup{p, x | n . we x ∈ A}, ∈ A subset K in the space we will call n x ∈ A}, p ∈ R . A subset K in the space R we will call n . A subset K in the space Rn we will call x ∈ A}, p ∈ R x ∈ A}, p ∈ R . A subset K in the space R we will call This work was This work was This work was Research (project This work was Research (project Research (project (project Research
supported by the Russian supported by the Russian supported by the Russian nos. 13-01-00295a). supported by the Russian nos. 13-01-00295a). nos. 13-01-00295a). 13-01-00295a). nos.
Foundation Foundation Foundation Foundation
for for for for
Basic Basic Basic Basic
cone if for any x ∈K and any λ > 0 the inclusion λx ∈K cone if for x K any λ > inclusion λx cone if Polar for any anycone x∈ ∈ to K and and any λK > 000⊂the the inclusion λx ∈ ∈K K cone for any x ∈ K and > the λx ∈ K holds.if the any coneλ Rnnninclusion is the the following following holds. Polar cone to the cone K ⊂ R is n is the following holds. Polar cone to the cone K ⊂ R holds. Polar cone to the cone K ⊂ R is the following subset subset subset subset 0 .. n 0 = n | p, x ≤ 0, K ∀x (1) ∈ R .. {p 0 n | p, x ≤ 0, K ∀x ∈ ∈ K}. K}. (1) {p ∈ R 0 = K (1) = {p ∈ R K = {p ∈ Rn || p, p, x x ≤ ≤ 0, 0, ∀x ∀x ∈ ∈ K}. K}. (1) The normed linear space of absolutely continuous functions The normed linear space of absolutely continuous functions The normed linear space continuous functions nof The normedby linear space absolutely continuous functions nof we denote AC(T, R ). absolutely The derivative of absolutely n we denote by AC(T, R The derivative of absolutely n ). we denote by AC(T, R ). The derivative of absolutely n we denote by AC(T, R ). The derivative of absolutely n ) may continuous function x(·) ∈ AC(T, R not n ) may not exist continuous function x(·) ∈ AC(T, exist n continuous function x(·) ∈Therefore AC(T, R Rdetermine may not not exist continuous AC(T, R )) may on a set set of of function measure x(·) zero. ∈ at exist each on a measure zero. Therefore determine at each on a set of measure zero. Therefore determine at each on a set of measure zero. Therefore determine at each point t of interval T a generalized (maybe multi-valued) point t of interval T a generalized (maybe multi-valued) point t of interval T a generalized (maybe multi-valued) point t of interval T a generalized (maybe multi-valued) derivative D(x(·), t), which coincides up to a factor with derivative D(x(·), t), which coincides up to a factor with derivative D(x(·), which coincides up to with derivative D(x(·), t), t), which coincides up to aa factor factor with classical derivative at the points where classical derivative classical derivative at the points where classical derivative classical derivative at the points where classical derivative classical derivative at the points where classical derivative exists. The The set set D(x(·), D(x(·), t) t) is is determined determined by by the the formula formula exists. .. set exists. The D(x(·), t) is determined by the formula exists. The D(x(·), t) = .. set D(x(·), t) is determined by the formula D(x(·), t) = D(x(·), t) = D(x(·), t) = x(t) − x(t − λ) . (2) x(t) − x(t − λ) (0) + B x(t) − x(t − λ) ε x(t) − x(t − λ) (0) .. (2) + B ε x(t) − x(t − λ) + λ + B (0) (2) ε (0) . (2) + B x(t) − x(t − λ) + λ ε ε>0 δ∈(0,t−t0 ) λ∈(0,δ) x(t) − x(t − λ) + λ ε>0 δ∈(0,t−t0 ) λ∈(0,δ) x(t) − x(t − λ) + λ ε>0 δ∈(0,t−t0 ) λ∈(0,δ) ε>0 δ∈(0,t−t0 ) λ∈(0,δ)
n n ), which is R We also define aa normed space AC ∞ (T, n R which is We also define normed space AC ∞ (T, n ), (T, R ), which We also define a normed space AC n ∞ (T, R ),continuous which is is We also define a space AC nnormed subset of AC(T, R ), consisting of absolutely ∞ n subset of AC(T, R ), consisting of absolutely continuous n ), consisting subset of AC(T, AC(T, R→ ofderivative absolutelysatisfies continuous n subset of R ), consisting absolutely continuous n , whose of functions f : T R the functions → R ,, nwhose derivative satisfies the functions ffff:::(·)T T∈ L → (T, RnnR whose derivative satisfies the functions T → R , nnwhose derivative satisfies the condition ), and the norm is determined ∞ (T, condition ff (·) ∈ L ), and the norm is determined ∞ (T, R . n condition (·) ∈ L R ), and the norm is determined ∞ (T, = . condition f (·) ∈ L R ), and the norm is determined n by the formula f ) + f . f (t ∞ AC∞ = Rn + f L∞ . by the formula f AC .. f (t00 ) L∞ by f )R AC∞ Rn n + f L∞ . ∞ = f (t0 by the the formula formula f n n AC∞ = f (t0 )R + f L∞ . n ) (F(Rn )) we denote the set of all (non-empty By P(R n n By P(R we denote the set of all (non-empty n) n )) By P(R )) (F(R (F(R )) denote of (non-empty n the By P(Rsubsets (F(R )) we we denote the set set(see of all all (non-empty n . Recall closed) of the space R Polovinkin and n closed) subsets of the space R . Recall (see and n . Recall (see Polovinkin closed) subsets of the space R Polovinkin and closed) subsets of the space R . Recall (see Polovinkin and Balashov (2007) §1.4 and Polovinkin (2014) §24), that the that the Balashov (2007) §1.4 and Polovinkin (2014) §24), Balashov (2007) §1.4 and Polovinkin (2014) §24), that the Balashov (2007) §1.4 and Polovinkin (2014) §24), that the lower tangent cone(also called: the simple tangent cone) to lower tangent cone(also called: the simple tangent cone) to lower tangent cone(also called: the simple tangent cone) to n called: the simple tangent cone) to lower tangent cone(also n at the point a ∈ A is the following subset the set A ⊂ R n at the point a ∈ A is the following subset the set A ⊂ R n the at the then point point a a∈ ∈A A −1 is the the following following subset subset .. at the set set A A⊂ ⊂R R is (A; a) a) = = {v ∈ ∈R Rnn || lim lim (v, (v, λ λ−1 (A − − a)) a)) = = 0}. 0}. (3) (3) T .. {v L (A; −1 (A T λ↓0 (v, TL L (A; (A; a) a) = = {v {v ∈ ∈R Rn || lim lim λ−1 (A (A − − a)) a)) = = 0}. 0}. (3) (3) T λ↓0 (v, λ L λ↓0 λ↓0
n n n → P(Rn ). Consider aa multi-valued mapping F :: T × R n n Consider multi-valued mapping F T × R P(R n → n ). Consider a multi-valued mapping F : T × R → P(R ). Consider a multi-valued mapping F : T × RF → P(Rset ). Recall that effective set of the mapping is the Recall that effective set of the mapping F is the set . Recall that effective set of the mapping F is the set . Recall that effective set of the mapping F is the set domF = {(t, x) | F (t, x) = ∅}. Graph of the mapping . domF = . {(t, domF = {(t, x) x) ||| F F (t, (t, x) x) = = ∅}. ∅}. Graph Graph of of the the mapping mapping domF = {(t, x) F (t, x) = ∅}. Graph of the mapping is the set is the set is the set .. is the set n n n × Rn | y ∈ F (t, x)}. graph F = x, y) ∈ T × R .. {(t, n graph F = {(t, x, y) ∈ T × R Rnn || yy ∈ F (t, x)}. n× graph F = {(t, x, y) ∈ T × R × graph F = {(t, x, y) ∈ T × R ×R R |y∈ ∈F F (t, (t, x)}. x)}. Also we call Also we call Also we .. Also graph we call callF (t, ·) = n n graph F (t, ·) = {(x, y) y) ∈ ∈R Rnnn × ×R Rnnn ||| yyy ∈ ∈F F (t, (t, x)}. x)}. .. {(x, graph graph F F (t, (t, ·) ·) = = {(x, {(x, y) y) ∈ ∈R R × ×R R |y∈ ∈F F (t, (t, x)}. x)}. n n n → P(Rn ) the Suppose that for aa mapping F :: T × R n Suppose that for mapping F T × R → P(R ) the n → P(Rn Suppose that for a mapping F : T × R Suppose that for a mapping Fis : defined T × R → P(Rn )) the the following differential inclusion following differential inclusion is defined following differential differential inclusion inclusion is is defined defined following
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150 150 150 150
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dx ∈ F (t, x), dt
t ∈ T.
(4)
Absolutely continuous function x(·) ∈ AC(T, Rn ), which . n is integrable branch of derivative x = dx dt : T → R the mapping F (·, x(·)) is called trajectory of differential inclusion (4) on segment T . By RT (F, C0 ) we denote the subset from AC(T, Rn ) of all the trajectories x(·) of differential inclusion (4) on the segment T provided that x(t0 ) ∈ C0 .
By R∞ T (F, C0 ) we denote the subset of RT (F, C0 ) consisting of all the trajectories x(·) from the space AC∞ (T, Rn ). 3. MEASURABLE-PSEUDO-LIPSCHITZ CONDITIONS
Hypothesis 1. Suppose that the mapping F : T × Rn → P(Rn ) such that there exist such a number δ > 0, . functions l(·) ∈ L∞ (T, R1+ ) and η(t) = δ(1 + 2l(t) exp((t − t0 )l(·)L∞ )), t ∈ T , and the closed subset W , satisfying inclusion: W ⊃ {(t, x) ∈ T × Rn | x − x (t) ≤ t ≤ 4δ exp((t − t0 )l(·)L∞ ) + 3 η(τ )dτ }, t0
that the following three conditions hold: ∀(t, x) ∈ W (5)
are non-empty, 2) for any function v(·) ∈ C(T, Rn ), for which graph v ⊂ W , the mapping t → G(t, v(t)) is measurable,
3) for any points (t, x1 ), (t, x2 ) ∈ W inclusions hold
Further we will need a result on the differentiation of the set of trajectories of differential inclusion with initial data from the paper Polovinkin (2013). Let the mapping F : T × Rn → F (Rn ) and the trajectory x (·) ∈ RT (F, C0 ) be defined. Define for any t ∈ T the following mapping . Ft : Rn → F(Rn ), where Ft (x) = F (t, x), and consider its lower multi-valued derivative at the point ( x(t), x (t)) ∈ n n n R × R in the direction u ∈ R (see Polovinkin (2013)) of the following type . . FL (t, u) = lim inf λ−1 (F (t, x (t) + λu) − x (t)) = λ↓0 . [λ−1 (F (t, x (t) + λu) − x (t)) + Bε (0)]. =
Similarly with notation RT (F, x0 ) for the set of trajectories of inclusion (4) by symbol RT (FL , u0 ) we denote the set of all solutions of Cauchy problems of the corresponding differential inclusion u (t) ∈ FL (t, u(t)),
t ∈ T,
(8)
with the initial condition u(t0 ) = u0 . From Polovinkin (2013) (see Theorems 3 and 4 and Corollary 5) and condition l(·) ∈ L∞ (T, R1+ ) one can obtain the following result. Theorem 1. Let subset C0 ⊂ Rn , mapping F : T × Rn → F(Rn ) and trajectory x (·) ∈ RT (F, C0 ) be given. Suppose that Hypothesis 1 is true at the neighborhood of trajectory x (·). Then for any u(·) ∈ RT (FL , TL (C0 , x (t0 ))), for which u (·) ∈ L∞ (T, Rn ), there exists such a number α > 0 that for any λ ∈ (0, α) there exist solution xλ (·) ∈ RT (F, C0 ) and a function o(λ, ·) from AC∞ (T, Rn ) such that (t) + λu(t) + o(λ, t), t ∈ T, xλ (t) = x −1 where lim λ o(λ, ·)AC∞ = 0. λ→+0
G(t, x1 ) ⊂ F (t, x2 ) + l(t)x1 − x2 B1 (0).
Definition 1. In the case when a mapping F : T × Rn → P(Rn ) satisfies Hypothesis 1, we say that the mapping F is called measurably-pseudo-Lipschitz in the neighborhood of trajectory x (·).
Hypothesis 2. Let the mapping F : T × Rn → P(Rn ) be measurable-pseudo-Lipschitz in the neighborhood of trajectory x (·) ∈ RT (F, x 0 ). Suppose that for subset of the following type . x) = G(t, F (t, x) ∩ ( x (t) + 3η(t)B1 (0)), ∀(t, x) ∈ W, (6) the next conditions hold:
2’) for any function v(·) ∈ C(T, R ) with graph v ⊂ W , v(t)) is measurable, the mapping t → G(t, n
3’) for any (t, x1 ), (t, x2 ) ∈ W inclusions hold x1 ) ⊂ F (t, x2 ) + l(t)x1 − x2 B1 (0). G(t,
4. DIFFERENTIATION OF THE SET OF TRAJECTORIES
ε>0 δ>0 λ∈(0,δ)
0 ) of differential incluFix some trajectory x (·) ∈ RT (F, x sion (4).
1) the subsets . G(t, x) = F (t, x) ∩ ( x (t) + η(t)B1 (0)),
151
(7)
Definition 2. In the case when a mapping F : T × Rn → P(Rn ) satisfies Hypothesis 2, we say that the mapping F is called strictly measurably-pseudo-Lipschitz in the neighborhood of trajectory x (·).
151
5. CONTINUOUS DEPENDENCE OF THE TRAJECTORIES ON THE INITIAL APPROXIMATIONS In the book Polovinkin (2014) there are some results on the continuous dependence of trajectories of the differential inclusion on initial approximations. Here are two results from §44 of this work. Let the mapping F : T × Rn → P(Rn ) satisfies Hypotheses 1 – 2 in the neighborhood of trajectory x (·) ∈ RT (F, x 0 ).
It means that the number δ ∈ (0, 1), functions l(·) and and the closed set W ⊂ T × Rn η(·), the mappings G, G are given.
Define a function m : T → R1+ of the following type . t m(t) = t0 l(τ ) dτ .
a) Consider an arbitrary function ρ0 (·) ∈ L1 (T, R1+ ) with ρ0 (t) ≤ η(t) for almost all t ∈ T . In the space AC(T, Rn ) define a subset . A(F, ρ0 (·)) = z(·) ∈ AC(T, Rn )z(t0 ) − x 0 ≤ δ,
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(z (t), F (t, z(t))) ≤ ρ0 (t), z (t) − x (t) ≤ η(t)
t∈T .
It is easy to show that the subset A(F, ρ0 (·)) is nonempty and graphs of all curves of this subset are contained in W . Moreover, for any z0 ∈ Bδ ( x0 ) there exists such z(·) ∈ A(F, ρ0 (·)) that z(t0 ) = z0 . . Theorem 2. Define a subset S0 = A(F, ρ0 (·)), where the function ρ0 (·) is determined above. Let the function d : Bδ ( x0 ) → Rn be continuous and such that d(z) − z < δ for all z ∈ Bδ ( x0 ). Define a finite set of trajectories . X = {wi (·)}Ii=1 ⊂ RT (G, Bδ ( x0 )) of the differential inclusion which is contained in the set S0 and d(wi (t0 )) = wi (t0 ) for all i ∈ 1, I. Then for any ε > 0 there exists a continuous mapping r: S0 → d(Bδ ( RT (G, x0 )), which satisfies: r(z(·))(t0 ) = d(z(t0 )) ∀z(·) ∈ S0 , r(wi (·))(·) = wi (·)) ∀i ∈ 1, I, t z (s) − d r(z(·))(s) ds ≤ ξε (t) ∀ t ∈ T, z(·) ∈ S0 , ds
t0
t
. b) We denote the standard simplex in R by Σ = k+1 . . γm = 1 . (9) = γ = (γ1 , . . . , γk+1 ) ∈ Rk+1 | γm ≥ 0, k
m=1
Theorem 3. Suppose that a subset C0 ⊂ Rn and a mapping F : T × Rn → F(Rn ), satisfying Hypothesis 2 near some trajectory x (·) ∈ RT (F, C0 ) are given. Let T0 be arbitrary regular tangent to the subset C0 at the point x (t0 ) cone. Suppose that finite set of trajectories um (·) ∈ R∞ T (FL , T0 ), m ∈ 1, k + 1 is given. Define k+1 . uγ (·) = γm um (·), γ ∈ Σk . m=1
(10)
Then there exists such a number α > 0 that for any λ ∈ (0, α) and for any γ ∈ Σk there exist a trajectory xλ,γ (·) ∈ RT (F, C0 ) and a function t → o(λ, γ, t) from AC∞ (T, Rn ) such that (t) + λuγ (t) + o(λ, γ, t), t ∈ T, xλ,γ (t) = x −1 lim max λ o(λ, γ, ·) AC∞ = 0, λ→+0
γ∈Σk
t0
t1 t
x∗ (s) ds ∈ K 0 (t),
a.e. t ∈ T. (14)
Consider the following extremal problem on the segment . T = [t0 , t1 ]:
t0
Let the following inclusion holds uγ (·) | γ ∈ Σk ⊂ RT (FL , T0 ).
7. TIME OPTIMUM PROBLEM
e−m(τ ) ρ0 (τ )dτ (1 + ε)) + ε. k+1
a.e.t ∈ T. Q(t, x) ∩ (γ(t)B1 (0)) = ∅, ∀x ∈ B1 (0), 0 0 Let K0 and K (t) be polar cones to the cones K0 and 0 K(t) respectively. Then the polar cone (R∞ T (F, K0 )) to ∞ ∗ ∗ the set RT (F, K0 ) consists of pairs of points b ∈ E and functions y ∗ (·) ∈ L1 (T, E ∗ ) such that for each pair there is a function x∗ (·) ∈ L1 (T, E ∗ ), for which the following inclusions hold t1 ∗ (13) b − x∗ (s) ds ∈ K00 ; x∗ (t), y ∗ (t) −
where
. ξε (t) = em(t)+ε (δ +
Theorem 4. Let K0 be a closed convex cone in a separable reflexive Banach space E. Let Q: T × E → F(E) be . such that Q(t, x) = {y ∈ E | (x, y) ∈ K(t)}, where the set K(t) is a closed convex cone in space E × E and it’s measurably time-dependent. Suppose that there is a function γ(·) ∈ L∞ (T, R1+ ) such that
(11)
(12)
besides, mappings γ → o(λ, γ, ·) from Σk to AC∞ (T, Rn ) are continuous. 6. THE POLAR OF A CONVEX PROCESS Due to Polovinkin (2012) one can calculate the polar cone to cone of trajectories of differential inclusion, the right part of which has a convex conical graph. 152
Minimize {t ∈ (t0 , t1 ] | x(·) ∈ RT (F, C0 ), x(t) ∈ C1 }. (15) where C0 , C1 ⊂ Rn are disjoint closed subsets.
In other words among all trajectories RT (F, C0 ) of differential inclusion (4), we are to find such a trajectory for which time τ of hitting the final subset C1 is minimal. . Suppose that such a time t = min τ and a trajectory x (·) exist. Then value t will be called optimal time, and x (·) solution of time optimum problem. Let the trajectory x (·) ∈ RT (F, C0 ) be solution of time optimum problem (15) and t - optimal time.
Formulate conditions on the mapping F in the problem (15). Suppose that Hypotheses 1 – 2 are satisfied. Hypothesis 3. Measurable mapping K : T → F(Rn × Rn ) is given, which values are closed convex cones satisfying inclusion K(t) ⊂ TL (graph F (t, ·); ( x(t), x (t))) a.e. t ∈ T, where TL (A, a) — the lower tangent cone to the set A at the point a ∈ A (see (3) ). Examples of mapping K(t) satisfying Hypothesis 3 are:
1) Clarke’s tangent cone to the set graph F (t, ·) at the point ( x(t), x (t)) (see §2.4 Clarke (1983));
2) asymptotic lower tangent cone to the set graph F (t, ·) at the point ( x(t), x (t)) (see §1.4 Polovinkin and Balashov (2007)).
Method of study of time optimum problem consists in the fact that using the optimal solution of the problem we investigate the set of trajectories of the differential inclusion in variations (8), which by virtue of Theorems 1 – 3 forms tangent cone to the set of trajectories of the original inclusion. Due to Theorem 4 we calculate the
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polar cone to cone of trajectories of differential inclusion in variations. Due to the optimality of a trajectory x (·) we show that this polar cone is in some sense a normal cone to a subset of trajectories of the original differential inclusion, from which we obtain necessary conditions optimality in the Lagrangian form. Theorem 5. Let the mapping F satisfy Hypotheses 1 – 3 at the neighborhood of x (·), also x (·) is the local solution of the time optimum problem (15) in the space AC(T, Rn ) and t is optimal time. Let K(t) from Hypothesis 3. Let K0 and K1 be some regular tangent cones to the sets C0 and C1 at the points x (t0 ) and x ( t) respectively. Then for any ξ ∈ D( x(·), t) (see (2)) there exists such a function t], Rn ), p(·) = 0 that the following formulae p(·) ∈ AC([t0 , hold: t) ∈ −K10 , (16) p(t0 ) ∈ K00 , p( p( t), ξ ≥ 0,
(p (t), p(t)) ∈ K 0 (t)
a.e. t ∈ [t0 , t],
(17)
(18)
where K00 , K10 , K 0 (t) – polar cones to the cones K0 , K1 , K(t) respectively (see (1)). 8. PROOF OF THE MAIN RESULT To prove Theorem 5 we transform original boundary value problem (15), defined in space Rn , to Cauchy problem in extended space Rn × Rn , in which for a point . y = (y 1 , y 2 ) ∈ Rn × Rn . we determine the norm y = max(y 1 , y 2 ). Define a . set C˜ = (C0 , C1 ) ⊂ Rn × Rn , a function . y (t) = ( x(t), x ( t)) (19)
: T × (Rn × Rn ) → P(Rn × Rn ) by and a mapping G formula . y) = G(t, {(v 1 , 0) ∈ Rn × Rn | v 1 ∈ G(t, y 1 )}. Define also a linear operator Λ : Rn × Rn → Rn of the following type . Λy = y 1 − y 2 . Due to optimality of the pair ( x(·), t) we have 0 = Λ y (t), 0 ∈ / Λ(A|t ) for all t ∈ [t0 , t), (20) where
. C)}. ˜ A|t = {y(t) ∈ (Rn × Rn ) | y(·) ∈ RT (G,
(21)
Determine mappings Q, P : T ×(Rn ×Rn )×R1 → P((Rn × Rn ) × R1 ) of the following type . Q(t, y, y 0 ) = y))}, (22) {(v, v 0 ) ∈ (Rn × Rn ) × R1 | v 0 ≥ (v, G(t, . P (t, y, y 0 ) = {(v, v 0 ) ∈ (Rn × Rn ) × R1 | v 0 ≥ (1 + l(·)L∞ ) ((y, v), R(t))}, . where R(t) =
{(y 1 , y 2 , v 1 , 0) ∈ Rn × Rn × Rn × Rn | (y 1 , v 1 ) ∈ K(t)}, and l(·) is from Hypothesis 1. It is easy to check that mappings Q and P are measurable in t ∈ T and for every t ∈ T are Lipschitz on y, y 0 in some neighborhood of the 153
153
point ( y (t), 0), the sets R(t), graph P (t, ·) are convex and closed. Denote . C = C˜ × {0} ⊂ (Rn × Rn ) × R1 , . D0 = K0 × K1 × {0} ⊂ Rn × Rn × R1 . In the same way as we did in §4, for almost every t ∈ T we introduce the notation of lower derivative of the mapping y (t), 0, y (t), 0) ∈ (y, y 0 ) → Q(t, y, y 0 ) at the point ( 0 graph Q(t, ·) in the direction (u, u ) ∈ (Rn × Rn ) × R1 of the following type . Q L (t, u, u0 ) = {(v, v 0 ) ∈ (Rn × Rn ) × R1 | (u, u0 , v, v 0 ) ∈ y (t), 0, y (t), 0))}. ∈ TL (graphQ(t, ·); ( t] and for any It is easy to show that for a.e. t ∈ [t0 , (u, u0 ) ∈ (Rn × Rn ) × R1 the following inclusion holds P (t, u, u0 ) ⊂ Q L (t, u, u0 ).
(23)
For an arbitrary vector ξ ∈ D( x(·), t) (see (2)) we define a set MΛ in the space Rn × R1 as follows . MΛ = {(τ ξ + Λu( t)) ∈ Rn × R1+ | τ ≤ 0, t), u0 ( (u(·), u0 (·)) ∈ RT (P, D0 ),
(u (·), u0 (·)) ∈ L∞ (T, (Rn × Rn ) × R1 )}.
(24)
Obviously that is convex cone. Prove that polar cone MΛ0 contains such point (q, q 0 ) ∈ Rn × R1 that q = 0. Assume the contrary. i.e. let MΛ0 ⊂ {(0, α) ∈ Rn × R1 | α ∈ R1 }.
(25)
Then MΛ00 = M Λ ⊃ {(x, 0) | x ∈ Rn }, whence by the convexity of the set MΛ we have the following inclusion (26) MΛ ⊃ Rn × (R1+ \ {0}). Based on the standard simplex Σn (see (9)), chose in Rn a simplex σ of the following type n+1 . . σ = {z = z(γ) | z(γ) = γi zi , γ ∈ Σn }, i=1
where zi ∈ Rn , i ∈ 1, n + 1 are the vertices of this simplex, such that 0 ∈ int σ. Let Γ be boundary of the simplex σ and . a = min{z | z ∈ Γ}. It follows from the inclusion 0 ∈ int σ that a > 0. Consider convex set . L = Λu( t) | (u(·), u0 (·)) ∈ RT (P, D0 ), (u (·), u0 (·)) ∈ a t) ≤ e−m( t)−1 . ∈ L∞ (T, (Rn × Rn ) × R1 ), u0 ( 8
From the definitions of the subset L, the inclusion (26) and the definition of subset MΛ (24) follows the equality (τ ξ + L) = Rn . (27) τ ≤0
From this equality and convexity of the set L follows monotonicity in τ ≤ 0 of the subsets τ ξ + L, i.e. for any τ1 , τ2 ∈ R1 , τ1 ≤ τ2 ≤ 0, the inclusion holds τ2 ξ + L ⊂ τ1 ξ + L. If ξ = 0 then the statement is obvious. If ξ = 0, then assume the contrary to prove the statement . Suppose that there exist such τ1 , τ2 ∈ R1 , τ1 < τ2 ≤ 0 that
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τ2 ξ+L τ1 ξ+L. Then there exists such point m0 ∈ L that . m0 ∈ / τ3 ξ + L, where τ3 = τ1 − τ2 . By the theorem on the separability of convex sets there exists p ∈ Rn , p = 0, such that p, m0 ≥ τ3 p, ξ + s(p, L). Since p, m0 ≤ s(p, L), then p, ξ ≥ 0 and s(p, L) < +∞. Therefore for any τ ≤ 0 we get that s(p, τ ξ+L) = τ p, ξ+s(p, L) ≤ s(p, L) < +∞, i.e. the following inclusion holds . (τ ξ + L) ⊂ H = {x ∈ Rn | p, x ≤ s(p, L) < +∞}, τ ≤0
and besides H = Rn , which contradicts the equality (27).
From monotonicity in τ ≤ 0 of the sets τ ξ + L, the equality (27) and the fact that the simplex σ is bounded it follows that there exists such τ0 ≥ 1 that the following inclusion holds σ ⊂ −τ0 ξ + L. Therefore there exist trajectories (vi (·), vi0 (·)) ∈ RT (P, D0 ), i ∈ 1, n + 1 with significantly bounded derivatives such that a t) ≤ e−m( t)−1 , −τ0 ξ + Λvi ( t) = zi , (28) vi0 ( 8 t where m(t) = t0 l(s) ds. For any γ ∈ Σn (see (9)) we define in AC(T, (Rn × Rn ) × R1 ) functions n+1 . γi · (vi (·), vi0 (·)), (vγ (·), vγ0 (·)) = i=1
γ ∈ Σn .
(29)
Due to the convexity of the subset RT (P, D0 ) we obtain inclusions (vγ (·), vγ0 (·)) ∈ RT (P, D0 ), ∀ γ ∈ Σn .
Hence and from the inclusion (23) by Theorem 3 there exists such a number δ0 > 0 that for all α ∈ (0, δ0 ) and 0 for all γ ∈ Σn there exist trajectories (yα,γ (·), yα,γ (·)) ∈ RT (Q, C) such that . yα,γ (t) = y (t) + αvγ (t) + o(t, α, γ), . 0 yα,γ (t) = αvγ0 (t) + o0 (t, α, γ), moreover lim max α−1 (o(·, α, γ), o0 (·, α, γ))AC = 0,
α→0 γ∈Σn
(30)
0 and functions γ → (yα,γ (·), yα,γ (·)) from Σn to AC(T, (Rn n 1 ×R )×R ) are continuous and uniformly bounded by some additive function for all α, γ. Therefore due to inequalities that follow from the definition (22) of mapping Q 0 yα,γ (t))) yα,γ (t) ≥ (y α,γ (t), G(t,
(31)
by Theorem 2 there exist δ1 ∈ (0, δ0 ) and trajectories yα,γ (t0 )), α ∈ (0, δ1 ) zα,γ (·) ∈ RT (G,
which continuously depend on γ ∈ Σn , moreover due to estimates from Theorem 2 the next inequalities are true t αa αa m( t)+1 0 yα,γ (s) ds + ≤ , zα,γ (t) − yα,γ (t) ≤ e 8 4 t0
t ∈ [t0 , t], α ∈ (0, δ1 ). In turn from the equality (30) there exists δ2 ∈ (0, δ1 ] such that αa a.e. t ∈ [t0 , t], ∀ α ∈ (0, δ2 ), γ ∈ Σn . o(t, α, γ) < 4
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Thus for all α ∈ (0, δ2 ) and for all γ ∈ Σn the function zα,γ (·) can be represented as follows zα,γ (t) = y (t) + αvγ (t) + αδ(t, α, γ), (32)
where functions γ → δ(·, α, γ) are continuous on Σn , and inequalities hold true a δ(·, α, γ)C < , ∀α ∈ (0, δ2 ), γ ∈ Σn . 2 Due to the definitions of the vector ξ ∈ D( x(·), t) (see (9)) and the function y (·) (see (19)) there exist . sequences λk → 0, β k = (βk , 0) ∈ Rn × Rn , βk → 0 such that y ( t) = y ( t − λk ) + αk ζ + αk β k , where . ζ = (ξ, 0) ∈ Rn × Rn and . αk = t) + λk . y ( t − λk ) − y ( Rewrite the last equality, substituting in it y ( t − λk ) from equality (32), where t = t − λk and α = αk /τ0 , τ0 is obtained in (28). Thus we have αk αk y ( t) + (−τ0 ζ + vγ ( t)) = z αk ,γ ( t − λk ) + ηk (γ), (33) τ 0 τ0 τ0 where the functions αk . ηk (γ) = τ0 βk + vγ (t) − vγ (t − λk ) − δ t − λk , ,γ τ0 are continuous on Σn and there exists such a number k0 that maxn ηk0 (γ) < a/2. γ∈Σ
Since γ = (γ1 , . . . , γn+1 ) ∈ Σn are barycentric coordinates on the simplex σ, then the function η: σ → E × E of . the following type η(z) = ηk0 (γ) (provided z = z(γ)) is continuous on σ. Consider a continuous function f : σ → Rn of the type: . f (z) = z − Λη(z). By the construction for any z ∈ Γ we have z ≥ a, i.e. for all z ∈ Γ the following inequality holds f (z) − z ≤ 2 max{ηk0 (γ) | γ ∈ Σn } < a ≤ z − 0. Hence by the theorem (scholium) in Sec. 4.1 in the book Lee and Markus (1972) the inclusion 0 ∈ f (σ) holds, that is there exists such a point z0 = z(γ 0 ) ∈ σ that f (z0 ) = 0. By virtue of equalities (28) and (29) provided γ 0 ∈ Σn we have an expression z(γ 0 ) = −τ0 ξ + Λvγ 0 ( t), from which the equality f (z0 ) = 0 takes form (34) t) = Ληk0 (γ 0 ). −τ0 ξ + Λvγ 0 ( Applying the operator Λ to both sides of equality (33) and reducing two terms due to equality (34), we obtain the following equality Λ y ( t) = Λz αk0 ,γ 0 ( t − λk0 ). τ0
Since Λ y ( t) = 0 (see (20)), then t − λk0 ) = 0, Λz αk0 0 ( τ0
,γ
that is there exists a trajectory of differential inclusion which hits the final set at the moment t − λk0 , which
IFAC CAO 2015 October 6-9, 2015. Garmisch-Partenkirchen, Evgenii S. Polovinkin et al. / IFAC-PapersOnLine 48-25 (2015) 150–155 Germany
contradicts the optimality of the pair ( x(·), t) (see (20), (21)).
Thus it is proved that assumption (25) was wrong and there exists a point (q, q 0 ) ∈ MΛ0 with q = 0, that is due to (24) for any τ ≤ 0 and for any (v(·), v 0 (·)) ∈ RT (P, D0 ) the inequality holds t) + q 0 v 0 ( t) ≤ 0. Λ∗ q, τ ζ + v(
(35)
Note that the cone graphP (t, ·) is convex and closed, measurably depends on t on [t0 , t1 ], and satisfies the conditions of Theorem 4. Choosing τ = 0 in (35) we obtain that the function η ∗ = (b∗ , y ∗ (·)), where b∗ = y ∗ (t) ≡ (Λ∗ q, q 0 ) ∈ (Rn × Rn ) × R1 , on the segment [t0 , t] belongs to the cone (R[t0 , t] (P, D0 ))0 , whence by Theorem 4 for the function η ∗ = (b∗ , y ∗ (·)) t], (Rn × Rn ) × R1 ), there exists function x∗ (·) ∈ L1 ([t0 , for which inclusions (13), (14) hold. Choosing a function (¯ p(·), p¯0 (·)) ∈ AC(T, (Rn × Rn ) × R1 ) of the type . (¯ p(t), p¯ (t)) = b∗ − 0
t
x∗ (s)ds,
t
from inclusions (14), (13) we obtain (¯ p (t), p¯0 (t), p¯(t), p¯0 (t)) ∈ {τ (u∗ , 0, v ∗ , −1) | τ ≥ 0,
t]; (u∗ , v ∗ ) ∈ (1 + l(·)∞ )(R0 (t) ∩ B1R (0))}, t ∈ [t0 , p( t), p¯0 ( t)) = (Λ∗ q, q 0 ). (¯ p(t0 ), p¯0 (t0 )) ∈ D00 , (¯ 4n
In (35) choosing a trajectory (v(·), v 0 (·)) = 0, we obtain Λ∗ q, ζ ≥ 0, that is ¯ p( t), ζ ≥ 0.
Finally, representing p¯(t) ∈ Rn × Rn in the form p¯(·) = (p1 (·), p2 (·)), where p1 (t), p2 (t) ∈ Rn , we get p1 (t0 ) ∈ K00 ,
p1 ( t), ξ ≥ 0;
p2 (t0 ) ∈ K10 ,
p2 ( t) = −p1 ( t) = q, 1 1 0 (p (t), p (t)) ∈ K (t), t ∈ [t0 , t]. . Thus supposing that p(·) = p1 (·), we obtain that p(·) = 0 (since p( t) = −q = 0) and the formulae (16)–(18) hold true. p2 (t) ≡ 0,
REFERENCES J.P. Aubin, Lipschitz behavior of solutions to convex minimization problems, Math. of Oper. Res. 9, pp. 87 – 111 (1984). Ph. D. Loewen and R. T. Rockafellar. Optimal control of unbounded differential inclusions, SIAM J. Control and Optimization. 32 (2), pp. 442 – 470 (1994). A.D. Ioffe, Existence and relaxation theorems for unbounded differential inclusions, J. Convex Anal. 13 (2), pp. 353 – 362 (2006). F.H. Clarke, Optimization and Nonsmoth Analysis, (J. Wiley, New York, 1983) E. S. Polovinkin and G. V. Smirnov, An approach to the differentiation of many-valued mappings, and necessary conditions for optimization of solutions of differential inclusions, Differential Equations 22 (1986), pp. 660 – 668. Russian original in Differen. Uravn. 22 (1986), pp. 944 - 954. E. S. Polovinkin and G. V. Smirnov, Time-optimum problem for differential inclusions, Differential equations 22 (1986), pp. 940 – 952. Russian original in Differen. Uravn. 22 (1986), pp. 1351 – 1365.
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E. S. Polovinkin, The properties of continuity and differentiation of solution sets of Lipschitzean differential inclusions, in Modeling, Estimation and Control of Systems with Uncertainty, Ed. by G. B. Di Masi, A. Gombani, and A. B. Kurzhansky (Birkh¨ auser, Boston, 1991), Prog. Syst. Control Theory 10, pp. 349 - 360. E. S. Polovinkin, Necessary conditions for optimization problems with differential inclusions, in Set-Valued Analysis and Differential Inclusions, Ed. by A. B. Kurzhanski and V. M. Veliov (Birkh¨ auser, Boston, 1993), Prog. Syst. Control Theory 16, pp. 157 - 170. E. S. Polovinkin, Necessary conditions for an optimization problem with a differential inclusion, Proc. Steklov Inst. Math. 211, pp. 350 - 361 (1995). E. S. Polovinkin and M. V. Balashov, Elements of Convex and Strongly Convex Analysis, 2nd ed. (Fizmatlit, Moscow, 2007) [in Russian]. E. S. Polovinkin, On the calculation of the polar cone of the solution set of a differential inclusion, Proc. Steklov Inst. Math. 278, pp. 169 - 178 (2012). E. S. Polovinkin, Differential Inclusions with Measurable PseudoLipschitz Right-Hand Side, Proc. Steklov Inst. Math. 283, pp. 116 - 135 (2013) E. S. Polovinkin, Set-Valued Analysis and Differential Inclusions, (Fizmatlit, Moscow, 2014) [in Russian]. E. B. Lee and L. Markus, Foundations of Optimal Control Theory, (Wiley, New York, 1967).