BV solutions of rate independent processes driven by impulsive controls⁎

17th IFAC Workshop on Control Applications of Optimization 17th IFAC Workshop on Control Applications of Optimization Yekaterinburg, Russia, 2018 of Optimization 17th IFAC Workshop onOctober Control 15-19, Applications Available at www.sciencedirect.com Yekaterinburg, Russia, 2018 of online 17th IFAC Workshop onOctober Control 15-19, Applications Optimization Yekaterinburg, Russia, October 15-19, 2018 Yekaterinburg, Russia, October 15-19, 2018

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IFAC PapersOnLine 51-32 (2018) 361–366

BV BV BV BV

solutions of rate independent processes solutions of rate independent processes solutions of rate independent processes  driven by impulsive controls solutions of rate independent processes driven by impulsive controls driven by impulsive controls  driven by impulsive controls ∗∗ Olga N. Samsonyuk ∗∗ Sergey A. Timoshin ∗∗

Olga N. Samsonyuk ∗∗∗ Sergey A. Timoshin ∗∗ ∗∗ Olga N. Samsonyuk ∗ Sergey A. Timoshin ∗∗ ∗∗ Olga N. Samsonyuk Sergey A. Timoshin ∗ ∗ Matrosov Institute for System Dynamics and Control Theory of SB Matrosov Institute for System Dynamics and Control Theory ∗ ∗ Matrosov InstituteRussia for System Dynamics and Control Theory of of SB SB RAS, Irkutsk, (e-mail: [email protected]) ∗ RAS, Irkutsk, Russia (e-mail: [email protected]) ∗∗Matrosov Institute for System Dynamics and Control Theory of SB RAS, Irkutsk, Russia (e-mail: [email protected]) Matrosov Institute for System Dynamics and Control Theory of ∗∗ Matrosov Institute for System Dynamics and Control Theory of SB SB ∗∗ RAS, Irkutsk, Russia (e-mail: [email protected]) ∗∗ RAS, Irkutsk, Russia (e-mail: [email protected]) Matrosov Institute for System Dynamics and Control Theory of SB ∗∗ RAS, Irkutsk, Russia (e-mail: [email protected]) Matrosov InstituteRussia for System Dynamics and Control Theory of SB RAS, Irkutsk, (e-mail: [email protected]) RAS, Irkutsk, Russia (e-mail: [email protected]) Abstract: Abstract: An An extension extension to to discontinuous discontinuous inputs inputs of of bounded bounded variation variation of of the the play play operator operator is is presented. The motivation for such an extension comes from optimal impulsive control problems Abstract: An extension to discontinuous inputs of bounded variation of the play operator is presented. The motivation for such an extension comes from optimal impulsive control problems Abstract: An motivation extension to discontinuous inputs ofvariational bounded variation of the play operator is presented. The forintroduced such an extension comes from optimal impulsive control problems with hysteresis. The latter is by means of inequalities modeling the action with hysteresis. The latter is introduced by meanscomes of variational inequalities modeling the action presented. The motivation forBV such an extension from optimal impulsive control problems of the play operator on the –inputs. Our extension is applied to the graph completions of with hysteresis. The latter is introduced by means of variational inequalities modeling the action of thehysteresis. play operator on the –inputs.by Our extension is applied to the graph completions of with Theitlatter is BV introduced means of control variational inequalities modeling the action BV –functions and allows us to study impulsive systems by measureof the play operator on the BV –inputs. Our extension is applied tocharacterized the graph completions of BV –functions and it allows us to study impulsive control systems characterized by measureof the play operator on the with BV –inputs. Our extension is applied to the approximation graph completions of driven differential equations nonlinear terms of hysteresis type. Some results BV –functions and it allows us to study impulsive control systems characterized by measuredriven differential equations with nonlinear terms of hysteresis type. Some approximation results BV –functions and it allows us to study impulsive control systems characterized by measuredriven differential equations with nonlinear terms of hysteresis type. Some approximation results for impulsive processes are provided. for impulsive processes are also also driven differential equations withprovided. nonlinear terms of hysteresis type. Some approximation results for impulsive processes are also provided. © 2018, IFAC (International of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. for impulsive processes areFederation also provided. Keywords: Hysteresis, impulsive and optimal Keywords: Hysteresis, impulsive and optimal controls, controls, measure-driven measure-driven differential differential equations. equations. Keywords: Hysteresis, impulsive and optimal controls, measure-driven differential equations. Keywords: Hysteresis, impulsive and optimal controls, measure-driven differential equations.     1. INTRODUCTION INTRODUCTION x(t) ˙˙ = ff t, x(t), y(t) G x(t) 1. + t, v(t), x(t) = t, x(t), y(t) + G t, x(t) v(t), x(a) x(a) = =x x0 ,, (4) (4) 1. INTRODUCTION x(t) ˙ = f t, x(t), y(t) + Gt, x(t)v(t), x(a) = x000 , (4) 1. INTRODUCTION x(t) ˙ f t, x(t), y(t) + G t, x(t) v(t), x(a) = x0 , (5) (4) q(t) for r Given aa time time interval interval T T = = [a, [a, b], b], we we denote denote by by W W 1,1 q(t) = = Ax(t) Ax(t) for all all tt ∈ ∈ T, T, (5) 1,1 (T, Rr ) Given 1,1 (T, Rr r) q(t) = Ax(t) for all t ∈ T, (5)   1,1 Given a time interval Tcontinuous = [a, b], wefunctions denote by W the space of on T Let (T,Z Rbe   all t ∈ T, q(t) F Ax(t) for (5) r) y(·) (6) the space of absolutely absolutely continuous functions onW T ..1,1 Let r= [a, b], we Given a time interval T denote by (T,Z Rbe ) y(·) = = F q(·), q(·), yy00 ,, (6) a closed convex set in R the following evolution the space of absolutely continuous on T . Let Z be r . Considerfunctions   y(·) = F q(·), y , (6) 0 a closed convex set in R . Consider the following evolution 0 r r following r Z be space of absolutely continuous functions on T . Let r athe closed convex set in R . Consider the evolution variational inequality. For qq :: T →R and R y(·) = F q(·),for y0 a. , e. t ∈ T, (6) v(t) ∈ K (7) r inequality. For Rrrr following and yy00 ∈ ∈evolution Rrrr with with v(t) ∈ K for a. e. t ∈ T, (7) rT →the avariational closed convex set in R . Consider q(a) − yy0 ∈ Z, find yy :: T that variational inequality. For→ qR :rTsuch →R and y00 ∈ Rr with v(t) ∈ K 1,1for a.ne. t ∈ T, 1,1 (7) r ∞ m r q(a) − ∈ Z, find T → R such that rT → R and y0 ∈ R with 0 where x ∈ W (T, R ), y ∈ W (T, R ), v ∈ L (T, R ), variational inequality. For q : r 1,1 n 1,1 r ∞ m   v(t) x ∈∈ KW 1,1for a. e.), ty∈∈T, (7) q(a)ρ−−y00q(t) ∈ Z, find y : T → R such that where (T, R W (T, R ), v ∈ L (T, R  n 1,1 r ∞ m 1,1 n ), y ∈ 1,1 (T, r ), ∞ (T, m ), r + ≤ where x∈ W (T, R W R v ∈ L R ), A is an (r × n) constant matrix. The set K is a closed q(a)ρ−−y0q(t) ∈ Z, find y(t) y˙˙ : T → + y(t), y(t), y(t) ≤0 0R such that n 1,1 Ther set K is ∞ a closed m A is anx ∈ (r W ×1,1 n)(T, constant matrix. mR ), y ∈ where W (T, R ), v ∈ L (T, R ), ρ − q(t) + y(t), y(t) ˙ ≤0 convex cone in R playing the role of the control conA is an (r × n) constant matrix. The set K is a closed m playing the role of the control conconvex in ρ − q(t) + y(t), y(t) ˙ for all ≤ 0 ∈ Z and a. e. t ∈ T, (1) straints m A is an cone (r × n) R constant matrix. The set K is a closed m set. Furthermore, y(·) is the solution of (1)–(3) and convex cone in R playing the role of the control confor all ρ ρ ∈ Z and a. e. t ∈ T, (1) straints m set. Furthermore, y(·)the is the solution (1)–(3)conand for all ρ ∈ Z and a. e. t ∈ T, (1) straints ncone r convex inRR playing role of that theof x R ,, yy0 Furthermore, ∈ such qq(1)–(3) yy0 ∈ Z y(·)points is the solution ofcontrol and nset. r are initial 0 ∈ 0 − q(t) − y(t) ∈ Z for all t ∈ T, (2) for all ρ ∈ Z and a. e. t ∈ T, (1) x ∈ R ∈ R are initial points such that − ∈ Z . n r 0 0 0 0 straints set. Furthermore, y(·) is the solution of (1)–(3) and r q(t) − y(t) ∈ Z for all t ∈ T, (2) with x00 ∈ qR0nn= ,. yAx 0 ∈0 .Rr are initial points such that q0 0 − y0 0 ∈ Z 0 q(t) − y(t) ∈ Z for all t ∈ T, (2) with q . = Ax . 0 0 x ∈ R , y ∈ R are initial points such that q − y ∈ Z 0 0 0. 0 0 y(a) = yy0 .. ∈ Z (3) q(t) − for all t ∈ T, (2) q00 = . Ax 0 y(a) =y(t) (3) with 0 The system (S) is considered under the following assumpwith q . = Ax 0 0 y(a) = y00 . (3) The system 1,1 r The system (S) (S) is is considered considered under under the the following following assumpassumpThe existence (T, R tions: = y0 . of (3) r) They(a) existence of a a unique unique solution solution yy ∈ ∈1,1W W 1,1 tions:system (S) is considered under the following assump1,1 (T, r) r Rr 1,1 The (output) to (1)–(3) for every input q ∈ W (T, R ) and The existence of a unique solution y ∈ (T, R ) 1,1W r tions: (output) every solution input q ∈y W∈1,1 (T,1,1R(T, 1,1W r ) and (H functions ff (t, x, y), G(t, x) are continuous in t r to (1)–(3) The existence of afor unique Rr ) tions: r 1 ) The yy(output) R a classical result (Brezis (1973), Moreau (1977)). (H The (t, x) are continuous in (1)–(3) for every input q ∈ W (T, R ) and r isto 0 ∈ 1,1 r ∈ R is a classical result (Brezis (1973), Moreau (1977)). (H111 ))locally The functions functions fcontinuous (t, x, x, y), y), G(t, G(t, x) y. are continuous in tt 0 r to (1)–(3) for every input q ∈ W and Lipschitz in x, In addition, there 1,1 rand (output) (T, R ) r yThe ∈ R is a classical result (Brezis (1973), Moreau (1977)). corresponding solution operator F : W and locally Lipschitzfcontinuous in x, y. In addition, there 0 1,1 (T, Rr ) × 0 (H1 )locally The functions (t, x, y), x) are in t r The corresponding solution operator F W R × exist two constants cc1 G(t, and cc2 y. such that Lipschitz continuous in x, Incontinuous addition, there r yR0rr ∈→R is1,1 a classical result (Brezis (1973), (1977)). 1,1 (T, The corresponding operator F ::Moreau W 1,1 (T, Rrrr ))the × and exist two positive positive constants and such that W (q, y ) →  y is the play operator for 1,1 (T, Rr ) : solution 0 1 2 and locally Lipschitz continuous in x, y. In addition, there 1,1 Rrr →corresponding W 1,1 (q, y0 ) →operator y is the play for 1,1 (T, Rr r ) : solution positive constants c11 and|G(t, c22 such that The F reader :operator W is(T, R )the × exist |f (t, (t,two x, y)| y)| ≤ cc1 (1 + |x| + |y|), x)| ≤ cc2 (1 + |x|), properties and applications of R (T, R is the the play operator for the 0 ) → exist two positive constants c1 and|G(t, c2 such that + |x| + |y|), x)| ≤ |f x, ≤ r → W 1,1and r ) : (q, y0 properties applications ofyywhich which the reader is referred referred 1 (1 R → W (T, R ) : (q, y ) →  is the play operator for the (t, x, y)| ≤ c (1 + |x| + |y|), |G(t, x)| ≤ c222 (1 (1n+ + |x|), |x|), |f 0 1 to Brokate and Sprekels (1996), Duvaut and Lions (1976), properties and applications of which the reader is referred 1 × Rrr .. (t, x, y) ∈ T × R n to Brokate and Sprekels (1996), Duvaut and Lions (1976), (1 + |x| + |y|), |G(t, x)| ≤ c (1 + |x|), |f (t, x, y)| ≤ c properties and applications of which the reader is referred 1 2 × (t, x, y) ∈ T × R n to Brokate and Sprekels (1996), Duvaut and Lions (1976), n×R Krej˘ c ´ ı (1991, 1996); Visintin (1994). This operator is rate Rrrr . (t, x, y) ∈ T × R Krej˘ c´ı (1991, 1996); Visintin (1994). This operator is rate We note that since the right-hand side of (4) is unbounded, n to Brokate and Sprekels (1996), Duvaut and Lions (1976), independent: × R . (t, x, y) ∈ T × R Krej˘ c ´ ı (1991, 1996); Visintin (1994). This operator is rate We that the right-hand side of unbounded, independent: We note note that since since the right-hand sidesystem of (4) (4) is is unbounded, Krej˘ c´ı (1991, 1996); Visintin (1994). This operator is rate the trajectories (x, y) of the control (S) may tend independent: the trajectories (x, y) of the control system (S) may tend We note that since the right-hand sidesystem ofConsequently, (4) is unbounded, ◦ ϕ, y ) = F(q, y ) ◦ ϕ 0 0 independent: F(q pointwise to discontinuous functions. the the trajectories (x, y) of the control (S) may tend F(q ◦ ϕ, y0 ) = F(q, y0 ) ◦ ϕ pointwise to discontinuous functions. Consequently, the the trajectories (x, y)inofgeneral, the functions. control system (S) may tend F(q ) = F(q, y ) ◦ ϕ 0 1,1 ◦ ϕ, ry0 0 0 tube of solutions is, not closed in the topology pointwise to discontinuous Consequently, the for any q ∈ W R any increasing surjective 1,1 (T, ry)0 )and F(q ◦ ϕ, = F(q, y ) ◦ ϕ tube of solutions is, in general, not closed in the topology 0 for any q ∈ W 1,1 pointwise to discontinuous functions. Consequently, the 1,1 (T, Rr r ) and any increasing surjective tube of solutions is, in general, not closed infaced the topology of uniform convergence. Hence, we may be with the Lipschitz reparametrization of time ϕ : T → T . for any q ∈ W (T, R ) and any increasing surjective uniform convergence. Hence, not we may beinfaced with the 1,1 r Lipschitz reparametrization of time ϕincreasing : T → T . surjective of tube of solutions is, in general, closed the topology for any q ∈ W (T, R ) and any situation when a sequence of controls {v } converges of uniform convergence. Hence, we may be faced with the k } converges to Lipschitz reparametrization of time ϕ : T → T . situation when a sequence of controls {v to k faced with the of uniform convergence. Hence, we may be This paper is devoted to the study of possible extensions of Lipschitz reparametrization of time ϕ : T → T . a Dirac delta function in the sense of distributions situation when a sequence of controls {v to k k } convergesand This paper is devoted to the study of possible extensions of a Dirac delta function in the sense of distributions and situation whenfunction a sequence ofofcontrols {v to} the play operator to discontinuous -inputs. Our motiThis paper is devoted to the study ofBV possible extensions of athe k }{xconverges corresponding sequences trajectories } and {y Dirac delta in the sense of distributions and k k the play operator to discontinuous BV -inputs. Our motiThis paper is devoted to the study ofBV possible extensions of the corresponding sequences of trajectories {x and {y k} k} a Dirac delta function in the sense of distributions and vation to consider these extensions originates in optimal the play operator to discontinuous -inputs. Our motithe corresponding sequences of trajectories {x } and {y tend pointwise functions. Optimization k problems k} k k vation consider these extensions originates in optimal the playto to discontinuous BV -inputs. Our motitend pointwise to to jump jump functions. Optimization vation tooperator consider these extensions originates in optimal the such corresponding sequences of trajectories {xk } problems and {ysoimpulsive control problems with hysteresis governed by k} for kind of control systems do not, in general, have tend pointwise to jump functions. Optimization problems impulsive control problems with hysteresis by such kind of to control systems do not, in general,problems have sovation to consider these extensions originatesgoverned in optimal impulsive control variational problems with hysteresis governed by for tend pointwise jump functions. Optimization rate independent inequalities. lutions the class of which is suchin control systemscontrols do not, in general, have sorate independent inequalities. impulsive control variational problems with hysteresis governed by for lutions inkind theof class of ordinary ordinary which is witnessed witnessed for suchfollowing kind of control systemscontrols do not, in general, have sorate independent variational inequalities. by the example. lutions in the class of ordinary controls which is witnessed We are concerned with the following control system (S): rate independent variational inequalities. by the following example. lutions in the class of ordinary controls which is witnessed We are concerned with the following control system (S): by the following example. We are concerned with the following control system (S): by the following example. Example We are concerned with the following control system (S): Example 1. 1. Let Let T T = =Z Z= = [−1, [−1, 1]. 1]. Consider Consider the the following following optimal control problem: Example 1. Let T = Z = [−1, 1]. Consider the following  The work is partially supported by the Russian Foundation for optimal control problem:  The work is partially supported by the Russian Foundation for Example 1. Let T = Z = [−1, 1]. Consider the following optimal control problem:  J(v) = (8) Basic Project no. 18-01-00026. TheResearch, work is partially supported by the Russian Foundation for optimal control problem: J(v) =V V (1) (1) + + ax ax4 (1) (1) → → inf inf (8) Basic Research, Project no. 18-01-00026.  TheResearch, work is partially supported by the Russian Foundation for J(v) = V (1) + ax444 (1) → inf (8) Basic Project no. 18-01-00026. J(v) = V (1) + ax (1) → inf (8) Basic Research, Project no. 18-01-00026. 4 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Copyright © 2018 IFAC 361 Copyright © under 2018 IFAC 361 Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 361 10.1016/j.ifacol.2018.11.410 Copyright © 2018 IFAC 361

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subject to the dynamics x˙ 1 (t) = v1 (t),

x1 (−1) = 0, (9)

x˙ 2 (t) = x1 (t)v2 (t),

x2 (−1) = 0, (10)

x˙ 3 (t) = y(t),

x3 (−1) = 1, (11) 

2

x˙ 4 (t) = x21 (t) + x22 (t) + x3 (t) − |t| , x4 (−1) = 0, (12)   ρ − q(t) + y(t) y(t) ˙ ≤0 for all ρ ∈ Z, (13) q(t) − y(t) ∈ Z for all t ∈ [−1, 1], y(−1) = −1.

(14)

Here, v ∈ L∞ (T, R2 ), q = x1 + x2 , a > 0 is a sufficiently large constant,  t   . V (t) = |v1 (s)| + |v2 (s)| ds, t ∈ [−1, 1]. −1

One can easily verify that Problem (8)–(14) does not have solutions for control functions v belonging to L∞ (T, R2 ). Moreover, for all feasible controls v we have . √ J(v) > J ∗ = 4 2 − 2. Furthermore, there exists a sequence {vk } such that J(vk ) → J ∗ as k → ∞. Hence, the infimum of the functional J over such feasible controls is J ∗ . The proof of these statements follows from using the functions ϕ1 (x1 , x2 , V ), ϕ2 (x1 , x2 , V ) defined as follows  x2 + (V + x1 )x1 , x1 ≤ −V /3,     x − (1/8)(V − x )2 , x ∈  − V /3, 0, 2 1 1 ϕ1 (x1 , x2 , V ) =   2  x − (1/8)(V + x ) , x  2 1 1 ∈ 0, V /3 ,   x2 − (V − x1 )x1 , x1 ≥ V /3, ϕ2 (x1 , x2 , V ) = ϕ1 (x1 , −x2 , V ). Indeed, the functions ϕ1 and ϕ2 have strongly monotone properties for the subsystem (9), (10) (see Dykhta and Samsonyuk (2015), and also Clarke et al. (1998), Vinter (2000)). Furthermore, the reachable set (with respect to x1 , x2 ) at a time instant t ∈ [−1, 1] is given by the set  .  R(V ) = (x1 , x2 ) | ϕ1 (x1 , x2 , V ) ≤ 0, ϕ2 (x1 , x2 , V ) ≤ 0 for V ≥ 0. Using this set and taking into account that a > 0 is sufficiently large, we conclude that J(v) > J ∗ for all feasible controls v.

Then, consider the sequence of controls {vk } such that    0, t ∈ − 1, −2/k ,    √      t ∈ − 2/k, −1/k ,   2k,   t ∈ − 1/k, 1/k , v1,k (t) = 0,  √      − 2k, t ∈ 1/k, 2/k ,       0, t ∈ 2/k, 1 ,    0, t ∈ − 1, −1/k ,    √     ( 2 − 1)k, t ∈ − 1/k, 0 , v2,k (t) = √    −( 2 − 1)k, t ∈ 0, 1/k ,       0, t ∈ 1/k, 1 . We see that {vk } → 0 in the sense of distributions and the corresponding sequence {Vk } is uniformly bounded. 362

  Let x1,k (·), x2,k (·), x3,k (·), x4,k (·), yk (·) be the solutions of (9)–(14) corresponding to {vk }. Directly we obtain J(vk ) → J ∗ as k → ∞. Hence, {vk } is a minimizing sequence of controls.

Next, we define the functions x ¯1 (·), x ¯2 (·) with the properties x ¯1√ (t) = 0, x ¯2 (t) =√0 for all t ∈ [−1, 1] \ {0} and x ¯1 (0) = 2, x ¯2 (0) = 2 − 2. Furthermore, let x ¯4 (t) ≡ 0,   −t, t ∈ [−1, 0), −1, t ∈ [−1, 0), y¯(t) = x ¯3 (t) = t, t ∈ [0, 1], 1, t ∈ [0, 1]. We have x1,k (t) → x ¯1 (t), x2,k (t) → x ¯2 (t), x3,k (t) → x ¯3 (t), x4,k (t) → x ¯4 (t), yk (t) → y¯(t) for all t ∈ [−1, 1]. It is quite reasonable then to consider the functions x ¯1 (·)– x ¯4 (·), and y¯(·) as a generalized solution for (9)–(14). The problem (8)–(14) relaxed accordingly constitutes the main object of our research in the present paper. It will be considered in the framework of optimal impulsive control problems. We note that this relaxation naturally raises the question of generalizing the play operator to discontinuous inputs with bounded total variation. We address this question in Sect. 2, 3. Note that there are several ways to extend the operator F defined by (1)–(3) when q ∈ BV (T, Rr ) (cf. Krej˘c´ı and Recupero (2014), Krej˘c´ı and Lauren¸cot (2002), Recupero (2011), Kopfova and Recupero (2016)). We observe, however, that the extensions in these references can be applied to the trajectory relaxation of the control system (S) only when the controls are scalar nonnegative (or nonpositive) functions. In the present paper, an alternative extension of the play operator suitable for K-valued controls is proposed. 2. THE PLAY OPERATOR AND ITS EXTENSIONS TO DISCONTINUOUS INPUTS In this section, we introduce some basic facts about the play operator and we present some of its possible extensions to discontinuous inputs (see Kopfova and Recupero (2016), Krej˘c´ı and Lauren¸cot (2002) and the references therein for more details). Let T = [a, b], H be a real Hilbert space with inner product ·, ·, and W 1,1 (T, H) be the space of absolutely continuous functions from T to H. By BV (T, H) we denote the space of H-valued functions of bounded variation (BV functions) and BVr (T, H) is the space of right continuous BV -functions on (a, b]. Given closed convex set Z ⊂ H, consider the evolution variational inequality: for q : T → H and y0 ∈ H with q(a) − y0 ∈ Z, find y : T → H such that   ρ − q(t) + y(t), y(t) ˙ ≤0 for all ρ ∈ Z

q(t) − y(t) ∈ Z

and a. e. t ∈ T,

for all t ∈ T

y(a) = y0 .

(15)

(16) (17)

1,1

1,1

Denote by F : W (T, H) × H → W (T, H) : (q, y0 ) → y the corresponding solution operator. Its continuity in the standard topology of W 1,1 (T, H) when H is separable and

IFAC CAO 2018 Olga N. Samsonyuk et al. / IFAC PapersOnLine 51-32 (2018) 361–366 Yekaterinburg, Russia, October 15-19, 2018

some regularity conditions are assumed is established in Krej˘c´ı (1991, 1996). We note that the relations (15)–(17) can be written in the form of the following differential   inclusion y˙ ∈ ∂IZ q − y , y(a) = y0 with q(0) − y0 ∈ Z. Here, IZ (·) is the indicator function of the set Z, i.e.,  0, z ∈ Z, IZ (z) = +∞ otherwise, and ∂IZ (·) is the subdifferential of the function IZ (·). Let H = R, Z = [−r, r], r > 0. Denote by F1 : W 1,1 (T, R) × R → W 1,1 (T, R) : (q, y0 ) → y the solution operator for the evolution variational inequality (15)–(17) in this case (scalar play operator). For a piecewise monotone input q ∈ W 1,1 (T, R) the output y = F1 (q, y0 ) is defined as follows (see Krasnoselskii and Pokrovskii (1989) for more details). We divide the interval T = [a, b] into subintervals inside which the function q stays monotone. Let a = t0 < t1 < . . . < tN = b be the corresponding partition of T and let ∆i = [ti−1 , ti ], i = 1, N . Then, the output y = F1 (q, y0 ) corresponding to q is defined by the rule:  following  y(0) = fr q(0), y0 ,   y(t) = fr q(t), y(ti−1 ) , t ∈ ∆i , i = 1, N ,   . where fr (q, y) = max q − r, min(q + r, y) .

There are several ways to extend the operator F in the case when q ∈ BV (T, H). Accordingly, one has several notions of a solution to (15)–(17) if input are discontinuous. In Recupero (2009a,b, 2011, 2015) one uses an approach based on the time-reparametrization. When q ∈ BVr (T, H) the time-reparametrization is the function b−a · V (t; q) ϕq (t) = V (b; q) . for V (t; q) = var q(·) being the total variation of q on the [a,t]

interval [a, t]. In this case, inequality (15) is replaced by  b   ρ(t) − q(t) + y(t), dy(t) ≤ 0 (18) a for all ρ ∈ BVr (T, H), where the integral is taken with respect to the LebesgueStieltjes measure dy. The continuity of the solution operator for (16)–(18) in the BV -norm topology is established in Kopfova and Recupero (2016). Recall that this topology is induced by the norm ||q||BV = ||q||∞ + V (b; q).

The play operator considered more generally on the space of regulated functions was treated in Krej˘c´ı and Lauren¸cot (2002); Krej˘c´ı and Liero (2009); Krej˘c´ı and Roche (2011). A regulated function is a function having both one-side finite limits at each point in its domain. The corresponding extension of the play operator admits a variational representation when the variational inequality (15) is replaced by the integral inequality  b   ρ(t) − q(t+) + y(t+), dy(t) ≤ 0 (19) a for all ρ ∈ Reg(T, Z), where Reg(T, Z) is the space of regulated Z-valued functions. The integral in (19) is understood in the sense of 363

363

Young (Krej˘c´ı and Lauren¸cot (2002)) or Kurzweil (Krej˘c´ı and Liero (2009), Krej˘c´ı and Roche (2011)). The continuity (with respect to the topology of uniform convergence) of the solution operator for (16), (17), (19) was proved in both Young and Kurzweil settings. The condition which guarantees the equivalence of the two extensions mentioned above is given in Krej˘c´ı and Recupero (2014). The extensions of the play operator relying on the relations (16)–(19) are not appropriate for relaxation of singular optimal control problems. This is illustrated in the following example. Example 2.

Consider the optimal control problem:  1  2 J(v) = x2 (t) − t dt → inf (20) 0

subject to the dynamics: x˙ 2 = y, x˙ 1 = v,   ρ − x1 + y y˙ ≤ 0

(21)

for all ρ ∈ [−1, 1] and a. e. t ∈ [0, 1],

x1 (0) = 0,

x2 (0) = 0,

y(0) = 0,

(22) (23)

y(t) − x1 (t) ∈ [−1, 1] for all t ∈ [0, 1], (24) ∞ where v ∈ L ([0, 1], R) and q = x1 . Define two sequences of controls     k, t ∈ [0, 2/k] , k, t ∈ [0, 1/k] , 2 1 vk = −k, t ∈ ( 2/k, 3/k] , vk =  0, t ∈ ( 1/k, 1] ,  0, t ∈ ( 3/k, 1] . These sequences converge to the Dirac function δ(t) concentrated at the point t = 0 in the sense of distributions. One can also show that the sequences of solutions {x11k }, {x21k } corresponding to {vk1 } and {vk2 } both converge pointwise to the function x ¯1 = χ(0,1] (t), where χ(0,1] (·) is the characteristic function of the interval (0, 1]. Moreover, for {vk1 } we have yk1 (t) → 0, x12k (t) → 0 for all t ∈ [0, 1]. The same output y¯ = 0 is obtained when (16)–(19) is applied to the input q = x ¯1 . Furthermore, for the sequence {vk2 } we have yk2 (t) → χ(0,1] (t), x22k (t) → t for all t ∈ [0, 1],

and it is easy to show that {vk2 } is a minimizing sequence for Problem (20)–(24). When relaxing Problem (20)–(24) ¯1 = χ(0,1] (t), x ¯2 = t, and y¯ = the limiting functions x χ(0,1] (t) should belong to the set of feasible solutions of the impulsive control system corresponding to (21)–(24). On the other hand, y¯ is not the output corresponding to the input q = x ¯1 for (16)–(19). When one wants to extend singular optimal control problems with hysteresis one needs to take into account the fact that the output of the hysteresis operator will depend on the way the input (function of bounded variation) is approximated by absolutely continuous functions. A similar problem which appears in the theory of impulsive control systems with vector measure controls and BV states was solved using the singular space-time transformation of the control system based on a discontinuous time reparametrization (Karamzin et al. (2014), Motta and Rampazzo (1995), Miller and Rubinovich (2003, 2013), Sesekin and Zavalishchin (1997)). Extending this idea an

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alternative extension of the play operator is proposed in the next section. 3. AN EXTENSION OF THE PLAY OPERATOR FOR GRAPH COMPLETIONS OF BV –FUNCTIONS In this section, an extension of the play operator suitable for relaxation of the control system (S) is proposed. The corresponding operator acts on the space of graph completions for functions of bounded variation. 3.1 Graph Completions for BV -Functions Recall that BVr ([a, b], Rn ) stands for the space of right continuous on (a, b] functions of bounded variation. Let x ∈ BVr ([a, b], Rn ). Denote by Sd (x) the set of jumps of x, i.e., . Sd (x) = {s ∈ [a, b] | x(s) − x(s−) = 0}. Given τ1 > 0, by a time-reparametrization we mean a nondecreasing Lipschitz continuous function η : [0, τ1 ] → [a, b] such that η(0) = a, η(τ1 ) = b. For a time reparametrization η(·), the pseudoinverse function θ : [a, b] → [0, τ1 ] is defined as follows θ(t) = inf{τ ∈ [0, τ1 ] | η(τ ) > t}, t ∈ (a, b], (25) θ(a) = 0. One can show that θ(·) is right continuous on (a, b] and increasing. In addition,   η θ(t) = t for all t ∈ [a, b]. For a time reparametrization η(·) and its pseudoinverse . . θ(·), let S η = Sd (θ), dηs = θ(s) − θ(s−). Definition 1. Given x ∈ BVr ([a, b], Rn ), the time reparametrization η(·) is said to be consistent with x(·) if Sd (x) ⊆ Sd (θ), where θ(·) is the pseudoinverse for η(·).   Denote by Tx [0, τ1 ], [a, b] the set of all time reparametrization which are consistent with x(·). Definition 2. Given x  ∈ BVr ([a,b], Rn ), a time reparametrization η ∈ Tx [0, τ1 ], [a, b] , and a family of Lipschitz continuous functions z s : [0, dηs ] → Rn , s ∈ η s s η S  , suchs that z (0) = x(s−), z (ds ) = x(s), the tuple x(·), {z (·)}s∈S η =: xη is called a graph completion corresponding to η(·) for x(·).   With xη = x(·), {z s (·)}s∈S η we associate a Lipschitz continuous function ξ : [0, τ1 ] → Rn such that    ∆s , (26) ξ(τ ) = x η(τ ) for all τ ∈ [0, τ1 ] \ s





s∈S η

ξ(τ ) = z τ − θ(s−) (27) for all τ ∈ ∆s and all s ∈ S η , . η where ∆s = [θ(s−), θ(s)], s ∈ S . One can show that   x(t) = ξ θ(t) for all t ∈ [a, b],   z s (τ ) = ξ τ + θ(s−) for all τ ∈ [0, dηs ] and all s ∈ S η . Observe that (ξ(·), η(·)) is a parametric representation of xη , which we denote in what follows by xη,τ . 364

3.2 Hysteresis Operator P First, we define the set of graph completions   r   q ∈ BVr ([a, b], R ),    .  BV gr [a, b], Rr = qη  η ∈ Tq [0, τ1 ], [a, b] , .   τ1 > 0

  Let q ∈ BVr ([a, b], Rr ), η ∈ Tq [0, τ1 ], [a, b] , and let θ(·) be the pseudoinverse for η(·) defined by (25), qη =   q(·), {zqs (·)}s∈S η be a graph completion corresponding to η(·) for q(·). As before, with qη we associate the function ζ : [0, τ1 ] → Rr defined by (26), (27) (with ξ, x, and z s replaced now by ζ, q, and zqs , respectively).   r Then, on the set BV gr [a, b], R we define the map P by   s η the rule: given qη = q(·), {zq (·)}s∈S , Z = [r1 , r2 ] ⊂ R, and y0 ∈ Rr such that q(a) − y0 ∈ Z, we put 

P(qη , y0 ) = yη ,  is defined as follows

y(·), {zys (·)}s∈S η

where yη =   y(t) = ν θ(t) for all t ∈ [a, b], (28)   zys (τ ) = ν τ + θ(s−) for all τ ∈ [0, dηs ], s ∈ S η , (29)

where ν = F(ζ, y0 ) and F : W 1,1 ([0, τ1 ], Rr ) × Rr → W 1,1 ([0, τ1 ], Rr ) is the play operator. One can show that y(·) ∈ BVr ([a, b], Rr ), yη is a graph correspond completion  ing to η(·) for y(·), and yη,τ = ν(·), η(·) . Note that the output ν(·) corresponding to given ζ(·) and y0 satisfies the variational inequality:   dν(τ ) ≤0 ρ − ζ(τ ) + ν(τ ) dτ for all ρ ∈ Z and a. e. τ ∈ [0, τ1 ], (30) ζ(t) − ν(t) ∈ Z

for all τ ∈ [0, τ1 ], ν(0) = y0 . (31)

Making use of (28), (29) and the integral form of the variational inequality (30):  τ1   ρ(τ ) − ζ(τ ) + ν(τ ) ν(τ ˙ )dτ ≤ 0 (32) 0 for all ρ ∈ BVr ([0, τ1 ], Z)

defined from (18), we can rewrite the relations (30), (31) in terms of graph completions. Then, we obtain a system of variational inequalities describing the components of yη , the output of P. The following proposition provides this system of variational inequalities.   gr r ∈ BV Proposition 1. Given q [a, b], R with qη = η   s r η q(·), {zq (·)}s∈S , y0 ∈ R with q(a) − y0 ∈ Z, and   yη = y(·), {zys (·)}s∈S η , the output of P corresponding to qη and y0 , the components of yη satisfy the following relations:    zys (dηs ) − y(s−) , t ∈ (a, b], y(t) = yc (t) + s≤t, s∈S η

y(a) = y0 . Here, yc (·) is a continuous solution of the variational inequality:

IFAC CAO 2018 Olga N. Samsonyuk et al. / IFAC PapersOnLine 51-32 (2018) 361–366 Yekaterinburg, Russia, October 15-19, 2018





b a

x(t) = x0 +

ρ(t) − q(t) + y(t), dyc (t) ≤ 0 for all ρ ∈ BVr (T ; Z),

q(t) − y(t) ∈ Z

η

for all t ∈ [a, b],

+

yc (a) = y0 ,

zys (·)

is a solution of the variational and for every s ∈ S inequality:   dzys (τ ) ρ − zqs (τ ) + zys (τ ), ≤0 dτ for all ρ ∈ Z and a. e. τ ∈ [0, dηs ], zqs (τ ) − zys (τ ) ∈ Z

for all τ ∈ [0, dηs ], zys (0) = y(s−).

4. IMPULSIVE CONTROL SYSTEMS . . m Define K1 = {v ∈ K | ||v|| = 1}, where ||v|| = j=1 |vj |, and denote by co A the convex hull of a set A. For a bounded Borel measure µ defined on T , denote by µc , |µc |, and Sd (µ) the continuous component in the Lebesgue decomposition of the measure µ, the total variation of µc , and the set on which the discrete component of µ is concentrated, i.e., . Sd (µ) = {s ∈ T | µ({s}) = 0}, respectively. A tuple

  π(µ) = µ, S, {ds , ωs (·)}s∈S

such that the following conditions are satisfied: i) µ is a K-valued bounded Borel measure on T , ii) the set S ⊂ T is at most countable subset of T , and Sd (µ) ⊆ S, iii) for every s ∈ S , ds ∈ R, ωs is a measurable function from [0, ds ] to co K1 such that  ds      ωs (τ )dτ = µ {s} , ds ≥ µ {s} , iv)



s∈S

0

ds < ∞

is called an impulsive control π(µ). Let W(T, K) denote the set of all impulsive controls π(µ). With every π(µ) ∈ W(T, K) we associate the following functions: V : T → R,

ηµ : [0, τ1,µ ] → [a, b]

  V (t) = |µc | [a, t] +

θµ (t) = t − a + V (t)



s∈S, s≤t

ds ,

t a



a

  f τ, x(τ ), y(τ ) dτ

 G τ, x(τ ) µc (dτ )

 

s∈S, s≤t

 z s (ds ) − x(s−) , t ∈ (a, b], x(a) = x0 , (33)

q(t) = Ax(t),   dz (τ ) = G s, z s (τ ) ωs (τ ), dτ s

t ∈ [a, b],

(34)

z s (0) = x(s−),

(35)

τ ∈ [0, ds ], s ∈ S, (36) zqs (τ ) = Az s (τ ),     y(·), {zys (·)}s∈S = P q(·), {zqs (·)}s∈S , y0 , (37)   (38) π(µ) = µ, S, {ds , ωs (·)}s∈S ∈ W(T, K), n r where x ∈ BVr (T, R ), q, y ∈ BVr (T, R ). Observe that     xηµ = x(·), {z s (·)}s∈S , qηµ = q(·), {zqs (·)}s∈S ,   yηµ = y(·), {zys (·)}s∈S are graph completions for x(·), q(·), and y(·), respectively. 5. APPROXIMATION RESULTS In this section we derive a relation between the control system (S) and the impulsive control system (D). To this aim, we introduce the notions of supplemented solutions for these two systems.   Let x(·), q(·), y(·) be the solution of (4)–(7) correspond. t ing to v ∈ L∞ (T, K) and let V (t) = a ||v(τ )||dτ . Consider the following pair of functions     XVac = x(·), V (·) , Qac V = q(·), V (·) ,   YVac = y(·), V (·) .   ac a supplemented solution We call the tuple XVac , Qac V , YV to (4)–(7), and let X ac denote the collection of all supplemented solutions to (4)–(7). Now, we give the notion of a supplemented solution for the impulsive control system (D). Consider an impulsive control π(µ) ∈ W(T, K) and let  xηµ , qηµ , yηµ , V (·) correspond to π(µ). Let set-valued functions XV , QV , YV be defined as follows:   (i) for t ∈ T \ S, XV (t) = x(t), V (t) ,     QV (t) = q(t), V (t) , YV (t) = y(t), V (t) ,

(ii) for t = s ∈ S,    XV (s) = z s (τ ), τ + V (s−) | τ ∈ [0, ds ] ,    QV (s) = zqs (τ ), τ + V (s−) | τ ∈ [0, ds ] ,    YV (s) = zys (τ ), τ + V (s−) | τ ∈ [0, ds ] . Then, a supplemented solution to (D) is the tuple   XV , QV , YV . Let X denote the collection of all supplemented solutions to (D).

. θµ : [a, b] → [0, τ1,µ ], where τ1,µ = b − a + V (b), such that

+



t

365

t ∈ (a, b], V (a) = 0,

for all t ∈ T,

Denote by comp(Rk ) the collection of all non-empty compact subsets from Rk . Let d(A, B) denote the Hausdorff distance between A, B ∈ comp(Rk ).

θµ (·) is the pseudoinverse function for ηµ (·). Consider the impulsive control system (D): 365

IFAC CAO 2018 366 Olga N. Samsonyuk et al. / IFAC PapersOnLine 51-32 (2018) 361–366 Yekaterinburg, Russia, October 15-19, 2018

For a set-valued function F : T → comp(Rk ), the graph of F is defined by the formula   graph F = (t, u) | t ∈ T, u ∈ F (t) . T   Lemma 2. Given XV , QV , YV ∈ X , there exists a sequence  ac  ac ac XV,k , Qac V,k , YV,k ⊂ X such that:   ac d graph XV , graph XV,k →0 as k → ∞, 

T

T

T

T

T

T

T

T

as j → ∞,

T

T

as j → ∞.

 d graph QV , graph Qac V,k → 0

as k → ∞,

  ac d graph YV , graph YV,k →0 as k → ∞. T T   ac ac Lemma 3. Given XV,k , Qac V,k , YV,k , a sequence of suppleac mented solutions from X such that the corresponding sequence {Vk (b)} is bounded,  there exist  a subsequence of  ac ac , Y , Q , Y and X ∈ X such that XV,kj , Qac V V V V,kj V,kj   ac as j → ∞, d graph XV , graph XV,kj → 0 

 d graph QV , graph Qac V,kj → 0   ac d graph YV , graph YV,k →0 j

Proofs of Lemmas 2, 3 are based on the space-time representation for the impulsive control system (D). These proofs follow closely the lines of the proof of a similar statements (for the case of the scalar play operator) from Samsonyuk and Timoshin (2017). The construction of this representation is based on the ideas developed in Dykhta and Samsonyuk (2015), Miller and Rubinovich (2003, 2013), Motta and Rampazzo (1995), Sesekin and Zavalishchin (1997). REFERENCES Brezis, H. (1973). Operateurs maximaus monotones et semi-groupes de contractions dans les espace de Hilbert. In North-Holland Mathematical Studies, volume 5. North-Holland Publishing Company, Amsterdam. Brokate, M. and Sprekels, J. (1996). Hysteresis and phase transitions. In Appl. Math. Sci., volume 121. SpringerVerlag, New York. Clarke, F., Ledyaev, Y., Stern, R., and Wolenski, P. (1998). Nonsmooth analysis and control theory. In Graduate Texts in Mathematics, volume 178. SpringerVerlag, New York. Duvaut, D. and Lions, J. (1976). Inequalities in Mechanics and Physics. Springer, Berlin. Dykhta, V. and Samsonyuk, O. (2015). HamiltonJacobi Inequalities and Variational Optimality Conditions. Irkutsk state university, Irkutsk, Russia. Karamzin, D., Oliveira, V., Pereira, F., and Silva, G. (2014). On some extension of optimal control theory. European Journal of Control, 20(6), 284–291. Kopfova, J. and Recupero, V. (2016). BV-norm continuity of sweeping processes driven by a set with constant shape. ArXiv: 1512.08711. Krasnoselskii, M. and Pokrovskii, A. (1989). Systems with Hysteresis. Springer-Verlag, Heidelberg. 366

Krej˘c´ı, P. (1991). Vector hysteresis models. European J. Appl. Math., 2, 281–292. Krej˘c´ı, P. (1996). Convexity and dissipation in hyperbolic equations. In Gakuto International Series Mathematical Science and Applications, volume 8. Gakk¯otosho, Tokyo. Krej˘c´ı, P. and Lauren¸cot, P. (2002). Generalized variational inequalities. J. Convex Anal., 9, 159–183. Krej˘c´ı, P. and Liero, M. (2009). Rate independent Kurzweil process. Appl. Math., 54, 117–145. Krej˘c´ı, P. and Recupero, V. (2014). Comparing BV solutions of rate independent processes. J. Convex. Anal., 21, 121–146. Krej˘c´ı, P. and Roche, T. (2011). Lipschitz continuous data dependence of sweeping processes in BV spaces. Disc. Cont. Dyn. Syst. Ser. B., 15, 637–650. Miller, B. and Rubinovich, E. (2003). Impulsive Controls in Continuous and Discrete-continuous Systems. Kluwer Academic Publishers, New York. Miller, B. and Rubinovich, E. (2013). Discontinuous solutions in the optimal control problems and their representation by singular space-time transformations. Autom. Remote Control, 74, 1969–2006. Moreau, J.J. (1977). Evolution problem associated with a moving convex set in a Hilbert space. J. Differential Eq., 26, 347–374. Motta, M. and Rampazzo, F. (1995). Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls. Differential Integral Equations, 8, 269–288. Recupero, V. (2009a). BV-extension of rate independent operators. Math. Nachr., 282. Recupero, V. (2009b). On a class of scalar variational inequalities with measure data. Applicable Analysis, 88(12), 1739–1753. Recupero, V. (2011). BV solutions of the rate independent inequalities. Ann. Scuola Norm. Sup. Pisa Cl. Sci., X(5), 269–315. Recupero, V. (2015). BV continuous sweeping processes. J. Differential Equation, 259, 4253–4272. Samsonyuk, O. and Timoshin, S. (2017). Optimal impulsive control problems with hysteresis. In Constructive Nonsmooth Analysis and Related Topics (dedicated to the Memory of V.F. Demyanov), 276–280. Sesekin, A. and Zavalishchin, S. (1997). Dynamic Impulse Systems: Theory and Applications. Kluwer Academic Publishers, Dordrecht. Vinter, R. (2000). Optimal Control. Birkhauser, Berlin, Germany. Visintin, A. (1994). Differential models of hysteresis. In Appl. Math. Sci., volume 111. Springer-Verlag, Berlin.