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Sign-Definiteness of the Integral Cost on a System of Linear Integral Equations of Volterra Type⁎
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 Yekaterinburg, Russia, October 15-19, 2018 Available online at www.sciencedirect.com 17th IFAC Workshop onOctober Control 15-19, Applications Yekaterinburg, Russia, 2018 of Optimization Yekaterinburg, Russia, October 15-19, 2018
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IFAC PapersOnLine 51-32 (2018) 424–427
Sign-Definiteness Sign-Definiteness of of the the Integral Integral Cost Cost on on Sign-Definiteness of the Integral Cost System of Linear Equations of Sign-Definiteness ofIntegral the Integral Cost on on System of Linear Integral Equations of System of Linear Integral Equations of Volterra Type System of Linear Integral Volterra Type Equations of Volterra Volterra Type Type Anastasia Samylovskaya ∗∗
a a a a
Anastasia Samylovskaya ∗∗ Anastasia Samylovskaya ∗ ∗ Anastasia Samylovskaya ∗ JSC RPE “Kvant”, Moscow, Russia (e-mail: JSC RPE “Kvant”, Moscow, Russia (e-mail: ∗ samylovskaya [email protected], [email protected]). RPE “Kvant”, Moscow, Russia (e-mail: [email protected]). ∗ JSC [email protected], samylovskaya JSC [email protected], RPE “Kvant”, Moscow, Russia (e-mail: samylovskaya [email protected]). samylovskaya [email protected], [email protected]). Abstract: Abstract: We We study study a a question question of of sign-definiteness sign-definiteness of of aa quadratic quadratic cost cost subject subject to to the the system system of linear integral equations which has no Legendre term, i.e. is totally degenerate. Therefore, Abstract: We study a question of sign-definiteness of a quadratic cost subject to the system of linear integral equations which has no Legendre term, i.e. is totally degenerate. Therefore, Abstract: Westudied study aby question ofhas sign-definiteness of a quadratic cost subjectapplying to the system it cannot be the clasical theory of quadratic However, some of linear integral equations which no Legendre term, i.e. forms. is totally degenerate. Therefore, it cannot be studied by the clasical theory of quadratic forms. However, applying some of linear integral equations which has no Legendre term, i.e.toforms. isreduce totally degenerate. Therefore, generalizations of the known Goh transform, it is possible the given functional to it cannot be studied by the clasical theory of quadratic However, applying some generalizations of the known Goh transform, it is possible to reduce the given functional to it cannot be studied by hence, theGoh clasical theory of quadratic forms. However, applying some a nondegenerate one, and to obtain new necessary conditions of its nonnegativity. Here generalizations of the known transform, it is possible to reduce the given functional to ageneralizations nondegenerateofone, and hence, to obtain new necessary conditions of its nonnegativity. Here the problem known Goh transform, itnecessary is possible to“classical” reduceof the given functional to we reduce the original to a form which differs from the one only by parameter a nondegenerate one, and hence, to obtain new conditions its nonnegativity. Here we reduce the original to atoform which differs from the “classical” onenonnegativity. only by parameter a nondegenerate one, problem and hence, obtain new necessary conditions of its Here which defines endpoint value of a new control, by using Goh transformation (Goh (1966)). Thus, we reduce the original problem to a form which differs from the “classical” one only by parameter which defines endpoint value ofto a new control, by usingfrom Gohthe transformation (Goh Thus, wenew reduce the original problem a form which differs “classical” one only(1966)). by optimality parameter a class of problems with reduced cost is investigated and corresponding necessary defines endpoint value of a new control, by using Goh transformation (Goh (1966)). Thus, awhich new class of problems with reduced cost is investigated and corresponding necessary optimality which defines endpoint value of a newcost control, by using Goh transformation (Goh (1966)). Thus, conditions are obtained. a new class of problems with reduced is investigated and corresponding necessary optimality conditions obtained.with reduced cost is investigated and corresponding necessary optimality a new class are of problems conditions are obtained. © 2018, IFAC of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. conditions are(International obtained. Federation Keywords: Keywords: Linear Linear Integral Integral Equation, Equation, Sign-Definiteness, Sign-Definiteness, Quadratic Quadratic Functional, Functional, Goh Goh Transformation Keywords: Linear Integral Equation, Sign-Definiteness, Quadratic Functional, Goh Transformation Keywords: Linear Integral Equation, Sign-Definiteness, Quadratic Functional, Goh Transformation Transformation 1. This question occurs in developming the second order 1. PROBLEM PROBLEM STATEMENT STATEMENT This question occurs in developming the second order optimality conditions control with 1. PROBLEM STATEMENT This question occurs of in optimal developming theproblems second order optimality conditions of control with 1. PROBLEM STATEMENT This question occurs in optimal developming theproblems second order integral equations of Volterra type, which are linear in optimality conditions of optimal control problems with Consider the following quadratic functional integral equations of Volterra type, which are linear in Consider the following quadratic functional optimality conditions of optimal control problems with control. integral equations of Volterra type, which are linear in Consider the following quadratic functional control. integral equations of Volterra type, which are linear in Consider the following quadratic functional control. Note that if the matrices A and B do not depend in control. Note that if the matrices A and B do not depend in Ω(x, the equation to Notefirst thatargument, if the matrices A (2) and is B equivalent do not depend in Ω(x, u) u) = = gg (x(0), (x(0), x(T x(T )) )) + + the first argument, equation (2) equivalent to ODE ODE T Note that if Bu, the matrices Acase andisis Bconsidered do not depend in Ω(x, u) = g (x(0), x(T )) + ˙x˙ = x Ax + and this by Goh T the first argument, equation (2) is equivalent to ODE =first Axargument, + Bu, and this case isis considered by ODE Goh Ω(x, u) = g (x(0), x(T )) + + ( Qx(t), x(t) + the equation (2) equivalent to T (1966) and Dmitruk (2008). Further, if the cost contains x ˙ = Ax Bu, and(2008). this case is considered by Goh + 0 T ( Qx(t), x(t) + (1966) and+ Dmitruk Further, if the cost contains x ˙ =“standard” Ax Bu,Legendre and(2008). thisterm case is considered by Goh + 0 ( Qx(t), x(t) + the (Ru, u) ,, then the (1966) and+Dmitruk Further, if the costfrom contains the “standard” Legendre term (Ru, u) then from the P (t)x(t), u(t) ) dt, (1) + 0 ( Qx(t),+ x(t) + (1966) andΩDmitruk (2008). Further, ifget the cost contains condition ≥ 0 on K one can easily the “classical” + P (t)x(t), u(t) ) dt, (1) the “standard” Legendre term (Ru, u) , then from the 0 condition Ω ≥ 0 on K one can easily get the “classical” + P (t)x(t), u(t) ) dt, (1) the “standard” Legendre term (Ru, u) , But then from the Legendre condition R(t) ≥ 0 for a.a. t. in our case condition Ω ≥ 0 on K one can easily get the “classical” + P (t)x(t), u(t) )function, dt, (1) Legendre condition R(t) ≥ can 0 foreasily a.a. t. But in“classical” our case where condition Ω ≥ 0 on K one get the where u(·) u(·) is is measurable measurable bounded bounded r−dimensional r−dimensional function, R(t) ≡ 0. Legendre R(t) ≥ 0 for a.a. t. But in our case R(t) ≡ 0. condition and x(·) is Lipschitz n−dimensional function T where u(·) measurable bounded r−dimensional function, Legendre and x(·) is is Lipschitz n−dimensional function on on [0, [0, T ]] that that R(t) ≡ 0. condition R(t) ≥ 0 for a.a. t. But in our case where u(·) is measurable bounded r−dimensional function, satisfy following linear equation of[0,Volterra Nevertheless, like and x(·)the is Lipschitz T ] that R(t) ≡ 0. satisfy followingn−dimensional linear integral integralfunction equationon like in in ODE ODE case, case, there there are are possible possible the the and x(·)the is Lipschitz n−dimensional function onof [0,Volterra T ] that Nevertheless, type further conditions, which are abscent in the general satisfy the following linear integral equation of Volterra Nevertheless, like in ODE case, there are possiblecase. the type further conditions, which are abscent in the general case. satisfy like in ODE there are general possiblecase. the type the following linear integral equation of Volterra Nevertheless, further conditions, which arecase, abscent in the Introduce the space type further conditions, which are abscent in the general case. Introduce the space t Introduce the space t t (A(t, τ )x(τ ) + B(t, τ )u(τ )) dτ, (2) Introduce the space x(t) = x + 0 x(t) = x0 + 0 t (A(t, τ )x(τ ) + B(t, τ )u(τ )) dτ, (2) W = Lipnn [0, T ] × Lrr∞ [0, T ] x(t) = x0 + 0 (A(t, τ )x(τ ) + B(t, τ )u(τ )) dτ, (2) W = Lipn [0, T ] × Lr∞ [0, T ] W = Lipn [0, T ] × L∞ [0, nT ] x(t) = is xa0quadratic + 0 (A(t, τ )x(τ ) +variables. B(t, τ )u(τ )) dτ, (2) and g(·, ·) form of 2n with elements W w = Lip (x, u), [0, nT ]is the space of [0, Twhere ] × Lr∞Lip and g(·, ·) is a quadratic form of 2n variables. 0 is the space of with elements w = (x, u), where Lip and g(·,matrix ·) is a quadratic form P of(t), 2n A(t, variables. n x(t), Lipschitz continuous n-vector functions and Lrr∞ of is All the functions Q(t), τ ) and B(t, τ ) are is the space with elements w = (x, u), where Lip andthe g(·,matrix ·) is a quadratic form P of(t), 2n A(t, variables. n x(t), Lipschitz continuous n-vector functions and L is All functions Q(t), τ ) and B(t, τ ) are the ∞ of isfunctions the space withspace elements w = (x, u), where Lip x(t), r of measurable bounded r-vector u(t). sufficiently smooth and uniformly bounded in t, τ . Lipschitz continuous n-vector functions and L All the matrix functions Q(t), P (t), A(t, τ ) and B(t, τ ) are ∞ r is the space of measurable bounded r-vector functions u(t). sufficiently smooth and uniformly bounded in t, τ . Lipschitz n-vector functions and Lu(t). All the matrix functions Q(t), P (t),bounded A(t, τ ) and ∞ is the space continuous of measurable bounded r-vectorx(t), functions sufficiently smooth and uniformly in t,B(t, τ . τ ) are We are the question sign-definiteness bounded r-vector functions u(t). sufficiently smoothin bounded in t, τ . of We are interested interested inand theuniformly question of of sign-definiteness of Ω Ω the space of measurable 2. on the of equation (2) satisfying the We are set interested in theto sign-definiteness Ω 2. THE THE CASE CASE OF OF ODE ODE on the set of solutions solutions toquestion equationof (2) satisfying also alsoof the We are interested in the question of sign-definiteness of Ω 2. THE CASE OF ODE following boundary conditions: on the set of solutions to equation (2) satisfying also the following conditions: 2. case THEwhen CASE on the setboundary of solutions to equation (2) satisfying also the Consider first the A OF and ODE B do not depend on following boundary conditions: Consider first the case when A and B do not depend on following boundary conditions: the first argument, i.e. A = A(τ ), B =do B(τ ). Consider first the case when A and not depend the on µ0 x(0) + µT x(T ) ≤ 0, the first argument, i.e. when A = A(τ ), BB B(τ ). Hence, Hence, Consider first the(2) case A to and B= do not depend the on integral equation is reduced the ODE µ0 x(0) + µT x(T ) ≤ 0, the first argument, i.e. A = A(τ ), B = B(τ ). Hence, the integral (2)i.e. is reduced (3) µ0 x(0) + µT x(T ) ≤ 0, the first equation argument, A = A(τto ), the B =ODE B(τ ). Hence, the (3) integral equation (2) is reduced to the ODE x(0) + µηη000x(0) x(T ))) = ≤ 0, 0, (3) integral equation (2) is reduced to the ODE x(0) + + µηηTTT x(T x(T = 0, (3) η0 x(0) + ηT x(T ) = 0, x x(0) + ηTofx(T ) = 0, x˙˙ = = Ax Ax + + Bu, Bu, x(0) x(0) = =x x00 .. where µ, η are someη0matrices corresponding dimensions. where µ, η are some matrices of corresponding dimensions. x ˙ = Ax + Bu, x(0) = x0 . where µ, η are some matrices of corresponding dimensions. The set L of all x˙the satisfying = pairs Ax + (x, Bu,u) x(0) = x0(2) . is The set L of all the pairs (x, u) satisfying (2) is a a subspace subspace where µ, η are some matrices of corresponding dimensions. This work is financially supported in part by the Russian Foundain W , and the set K of all the pairs (x, u) satisfying (3) The set L of all the pairs (x, u) satisfying (2) is a subspace This work is financially supported in part by the Russian Foundain W , and the set K of all the pairs (x, u) satisfying (3) is is The set L of all theK pairs (x, u)pairs satisfying (2) is a subspace a finite-faced cone in L. tion forwork BasicisResearch, projects no. in 16-01-00585 and 18-31-00091. in W , and the set of all the (x, u) satisfying is This financially supported part by the Russian Foundaainfinite-faced cone in of L.all the pairs (x, u) satisfying (3) tion for Basic Research, projects no. 16-01-00585 and 18-31-00091. W , and the set K (3) is This financiallyprojects supported part by theand Russian Foundaa finite-faced cone in L. tion forwork BasicisResearch, no. in 16-01-00585 18-31-00091. a finite-faced cone in L. tion for Basic Research, projects no. 16-01-00585 and 18-31-00091. 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2018 IFAC 424 Copyright 2018 IFAC 424 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 424 10.1016/j.ifacol.2018.11.421 Copyright © 2018 IFAC 424
IFAC CAO 2018 Anastasia Samylovskaya / IFAC PapersOnLine 51-32 (2018) 424–427 Yekaterinburg, Russia, October 15-19, 2018
As is known, in this case one can make a change of variables, in terms of which the quadratic form will contain the Legendre term with respect to some new control. First, introduce new state variables ξ, y satisfying system
y˙ = u,
y(0) = 0,
Introduce the matrix C(t) = B (t, t) and a new state variable ξ(t) = x(t) − C(t)y(t), where we still have y˙ = u, y(0) = 0,
ξ = x − Bu.
x(0) = ξ(0) = x0 = ξ0 .
The passage x −→ (ξ, y) is called Goh transform. Thus, we get
425
Then we substitute x = ξ + Cy to (2) and obtain the expression
ξ˙ = Aξ + AB − B˙ y. Rewriting Ω in new variables (ξ, y, u), we get the following expression:
ξ(t) = ξ0 +
t
(A(t, τ )ξ(τ ) + D(t, τ )y(τ )) dτ,
(5)
0
where Ω(ξ, y, u) = g1 (ξ(0), ξ(T ), y(T )) + T + ( Q1 ξ, ξ + P1 ξ, y + 0
+ Gy, u + Ry, y) dt,
(4)
where Q1 , P1 , R are some matrices, and g1 is a quadratic form of 2n + r arguments. Represent G in form of G = S + V , where S is symmetric, and V is skew-symmetric matrices. Using the equality ˙ y) + 2(Sy, y), (Sy, y)˙ = (Sy, ˙ we obtain T 0
D(t, τ ) = A(t, τ )C(t) − Bτ (t, τ ). As we can see, (5) is quite similar to (2). Thus, from the initial pair of variables (x(t), u(t)) we come to a triple (ξ(t), y(t), u(t)). Rewriting Ω in these new terms, we receive the following expression: Ω(ξ, y, u) = g1 (ξ(0), ξ(T ), y(T )) + T + ( Qξ, ξ + P1 ξ, y + 0
+ P1 ξ, u + R1 y, y + Gy, u) dt.
1 1 T ˙ (Sy, y)dt ˙ = Sy, y − (Sy, y)dt, 2 2 0 T
and all these summends ”fit in” well to the form of (4) without u. So, the only part of G which remains in (4) is the skewsymmetric part V . A key fact (see Goh (1966) and Dmitruk and Shishov (2010)) is that the nonnegativity of Ω on K implies that V (t) ≡ 0 on [0, T ]. So, we now can take y as a new control belonging to Lr2 [0, T ] instead of being Lipschitz-continuous, and one can show that this extension will not affect the nonnegativity of Ω. Therefore, Ω ≥ 0 on K implies the Legendre condition R(t) ≥ 0 on [0, T ], which is called, in our situation, the Legendre-Goh condition. Note also that instead of y(T ) one should take a parameter h ∈ Rr , for which relations (3) fulfill after substitution x(t) = ξ(t) + B(t)h. Thus, we have a situation that differs from the “classical” one only by the presence of parameter h. The signdefiniteness of such a degenerated functional involving the parameter h is studied in Dmitruk and Shishov (2010). 3. THE GENERAL CASE
Our nearest goal is to exclude the control u from Ω. Consider the term P1 ξ, u separately and obtain
T
0
Clearly, the first two terms ”fit in” well to the form of (4). Let us then consider the last one. In view of (5), ξ˙ = A(t, t)ξ(t) + D(t, t)y(t) + t + At (t, τ )ξ(τ ) + Dt (t, τ )y(τ )dτ. 0
Introduce one more state variable t At (t, τ )ξ(τ ) + Dt (t, τ )y(τ )dτ. z(t) = 0
Notice that z(0) = 0. Then ˙ = A(t)ξ(t) + D(t)y(t) + z(t), ξ(t) and so we obtain
0
Let us try to act in a way similar to the one used in the case of ODE. 425
P1 (t)ξ(t), y(t) ˙ = P1 (T )ξ(T ), y(T ) − T ˙ − P˙1 (t)ξ(t), y(t) + P1 ξ(t), y(t) dt.
0
T
˙ P1 ξ(t), y(t) dt =
0
T
P1 z(t), y(t) +
+ P2 Aξ(t), y(t) + P3 Dy(t), y(t) dt.
IFAC CAO 2018 426 Anastasia Samylovskaya / IFAC PapersOnLine 51-32 (2018) 424–427 Yekaterinburg, Russia, October 15-19, 2018
Thus, instead of one phase variable x(t) we now have a triple (ξ(t), y(t), z(t)). Herewith the cost will take the form Ω(ξ, y, z, u) = g2 (ξ(0), ξ(T ), y(T )) + T + Q(t)ξ(t), ξ(t) + P3 (t)ξ(t), y(t) + 0
+ P1 (t)y(t), z(t) + R1 (t)y(t), y(t) +
is dense in K.
+ G(t)y(t), u(t) dt. Notice that ξ and z satisfy integral equations of the same type and introduce ρ = (ξ, z). Hence, we have
ρ(t) = ρ(0) +
t 0
Γ(t, τ )ρ(τ ) + Λ(t, τ )y(τ ) dτ,
In the proof, we use the following property (Dmitruk and Shishov (2010)). Lemma 3.1. (Density Lemma). Let L be a locally convex space, K be a finite-faced cone in L, and L ⊂ L be a linear manifold dense in L. Then = K ∩ L K
(6)
where Γ and Λ are some new matrices, the specific form of which is not important for our purposes. Rewrite Ω in terms of (ρ, y, u):
Since we got rid of u in Ω, hence Theorem 3.1 yields the following Corollary 3.1. (Goh-Legendre condition) If Ω ≥ 0 on [0, T ], then R(t) ≥ 0 on [0, T ]. Finally, by the analogy with Section 2, one can show that the space y ∈ C can be extended to y ∈ L2 with the change of the terminal value y(T ) by an arbitrary vector h ∈ Rr . Then we have Ω(ρ, y, h) = g˜ (ρ(0), ρ(T ), h) + T Q(t)ρ(t), ρ(t) + P (t)ρ(t), y(t) + + 0
Ω(ρ, y, u) = g˜ (ρ(0), ρ(T ), y(T )) + T ˜ + Q(t)ρ(t), ρ(t) + P˜ (t)ρ(t), y(t) + 0 + R(t)y(t), y(t) + G(t)y(t), u(t) dt,
(7)
+ R(t)y(t), y(t) dt.
and we obtain one more condition concerning the parameter h: Corollary 3.2. For any pair (ρ, y) satisfying (6) the following should hold: inf Ω(ρ, y, h) ≥ 0,
By the analogy with the above, the matrix G(t) can be reduced to a skew-symmetric matrix V (t), so
h
where inf is taken over all h ∈ Rr satisfying the constraints Ω(ρ, y, u) = g˜ (ρ(0), ρ(T ), y(T )) + T + Q(t)ρ(t), ρ(t) + P (t)ρ(t), y(t) + 0
+ R(t)y(t), y(t) + V (t)y(t), u(t) dt.
Introduce the space
with elements
(8)
= C n × C r × Lr W ∞
σ = (ρ, y, u) and rewrite (3) in terms of our current variables. Then we obtain
M0 ρ(0) + MT ρ(T ) + My y(T ) ≤ 0, N0 ρ(0) + NT ρ(T ) + Ny y(T ) = 0,
(9)
where M, N are some matrices of corresponding dimensions. The set L of all the pairs (ρ, y) satisfying (6) is of all the pairs (ρ, y) a subspace in W , and the set K satisfying (9) is a finite-faced cone in W . Then the following statement takes place. then V (t) ≡ 0. Theorem 3.1. If Ω ≥ 0 on K,
426
M0 ρ(0) + MT ρ(T ) + My h ≤ 0, N0 ρ(0) + NT ρ(T ) + Ny h = 0. 4. CONCLUSION
Here we studied a question of sign-definiteness of a quadratic cost subject to the system of linear integral equations which has no Legendre term, i.e. is totally degenerate. Therefore, it cannot be studied by the clasical theory of quadratic forms. However, applying some generalizations of the known Goh transform, it is possible to reduce the given functional to a nondegenerate one, and hence, to obtain new necessary conditions of its nonnegativity. We reduced the original problem to a form which differs from the “classical” one only by parameter which defines endpoint value of a new control, by using Goh transformation (Goh (1966)). Thus, a new class of problems with reduced cost is investigated and corresponding necessary optimality conditions are obtained. ACKNOWLEDGEMENTS The author thanks Prof. Andrei V. Dmitruk for valuable remarks and discussions.
IFAC CAO 2018 Anastasia Samylovskaya / IFAC PapersOnLine 51-32 (2018) 424–427 Yekaterinburg, Russia, October 15-19, 2018
REFERENCES Dmitruk, A. (2008). Jacobi type conditions for singular extremals. Control and Cybernetics, 37(2), 285–306. Dmitruk, A. and Shishov, K. (2010). Analysis of a quadratic functional with a partly singular legendre condition. Moscow University Computational Mathematics and Cybernetics, 34(1), 16–25. Goh, B. (1966). Necessary conditions for singular extremals involving multiple control variable. SIAM J. Control, 4(2), 716–731.
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IFAC PapersOnLine 51-32 (2018) 424–427
Sign-Definiteness Sign-Definiteness of of the the Integral Integral Cost Cost on on Sign-Definiteness of the Integral Cost System of Linear Equations of Sign-Definiteness ofIntegral the Integral Cost on on System of Linear Integral Equations of System of Linear Integral Equations of Volterra Type System of Linear Integral Volterra Type Equations of Volterra Volterra Type Type Anastasia Samylovskaya ∗∗
a a a a
Anastasia Samylovskaya ∗∗ Anastasia Samylovskaya ∗ ∗ Anastasia Samylovskaya ∗ JSC RPE “Kvant”, Moscow, Russia (e-mail: JSC RPE “Kvant”, Moscow, Russia (e-mail: ∗ samylovskaya [email protected], [email protected]). RPE “Kvant”, Moscow, Russia (e-mail: [email protected]). ∗ JSC [email protected], samylovskaya JSC [email protected], RPE “Kvant”, Moscow, Russia (e-mail: samylovskaya [email protected]). samylovskaya [email protected], [email protected]). Abstract: Abstract: We We study study a a question question of of sign-definiteness sign-definiteness of of aa quadratic quadratic cost cost subject subject to to the the system system of linear integral equations which has no Legendre term, i.e. is totally degenerate. Therefore, Abstract: We study a question of sign-definiteness of a quadratic cost subject to the system of linear integral equations which has no Legendre term, i.e. is totally degenerate. Therefore, Abstract: Westudied study aby question ofhas sign-definiteness of a quadratic cost subjectapplying to the system it cannot be the clasical theory of quadratic However, some of linear integral equations which no Legendre term, i.e. forms. is totally degenerate. Therefore, it cannot be studied by the clasical theory of quadratic forms. However, applying some of linear integral equations which has no Legendre term, i.e.toforms. isreduce totally degenerate. Therefore, generalizations of the known Goh transform, it is possible the given functional to it cannot be studied by the clasical theory of quadratic However, applying some generalizations of the known Goh transform, it is possible to reduce the given functional to it cannot be studied by hence, theGoh clasical theory of quadratic forms. However, applying some a nondegenerate one, and to obtain new necessary conditions of its nonnegativity. Here generalizations of the known transform, it is possible to reduce the given functional to ageneralizations nondegenerateofone, and hence, to obtain new necessary conditions of its nonnegativity. Here the problem known Goh transform, itnecessary is possible to“classical” reduceof the given functional to we reduce the original to a form which differs from the one only by parameter a nondegenerate one, and hence, to obtain new conditions its nonnegativity. Here we reduce the original to atoform which differs from the “classical” onenonnegativity. only by parameter a nondegenerate one, problem and hence, obtain new necessary conditions of its Here which defines endpoint value of a new control, by using Goh transformation (Goh (1966)). Thus, we reduce the original problem to a form which differs from the “classical” one only by parameter which defines endpoint value ofto a new control, by usingfrom Gohthe transformation (Goh Thus, wenew reduce the original problem a form which differs “classical” one only(1966)). by optimality parameter a class of problems with reduced cost is investigated and corresponding necessary defines endpoint value of a new control, by using Goh transformation (Goh (1966)). Thus, awhich new class of problems with reduced cost is investigated and corresponding necessary optimality which defines endpoint value of a newcost control, by using Goh transformation (Goh (1966)). Thus, conditions are obtained. a new class of problems with reduced is investigated and corresponding necessary optimality conditions obtained.with reduced cost is investigated and corresponding necessary optimality a new class are of problems conditions are obtained. © 2018, IFAC of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. conditions are(International obtained. Federation Keywords: Keywords: Linear Linear Integral Integral Equation, Equation, Sign-Definiteness, Sign-Definiteness, Quadratic Quadratic Functional, Functional, Goh Goh Transformation Keywords: Linear Integral Equation, Sign-Definiteness, Quadratic Functional, Goh Transformation Keywords: Linear Integral Equation, Sign-Definiteness, Quadratic Functional, Goh Transformation Transformation 1. This question occurs in developming the second order 1. PROBLEM PROBLEM STATEMENT STATEMENT This question occurs in developming the second order optimality conditions control with 1. PROBLEM STATEMENT This question occurs of in optimal developming theproblems second order optimality conditions of control with 1. PROBLEM STATEMENT This question occurs in optimal developming theproblems second order integral equations of Volterra type, which are linear in optimality conditions of optimal control problems with Consider the following quadratic functional integral equations of Volterra type, which are linear in Consider the following quadratic functional optimality conditions of optimal control problems with control. integral equations of Volterra type, which are linear in Consider the following quadratic functional control. integral equations of Volterra type, which are linear in Consider the following quadratic functional control. Note that if the matrices A and B do not depend in control. Note that if the matrices A and B do not depend in Ω(x, the equation to Notefirst thatargument, if the matrices A (2) and is B equivalent do not depend in Ω(x, u) u) = = gg (x(0), (x(0), x(T x(T )) )) + + the first argument, equation (2) equivalent to ODE ODE T Note that if Bu, the matrices Acase andisis Bconsidered do not depend in Ω(x, u) = g (x(0), x(T )) + ˙x˙ = x Ax + and this by Goh T the first argument, equation (2) is equivalent to ODE =first Axargument, + Bu, and this case isis considered by ODE Goh Ω(x, u) = g (x(0), x(T )) + + ( Qx(t), x(t) + the equation (2) equivalent to T (1966) and Dmitruk (2008). Further, if the cost contains x ˙ = Ax Bu, and(2008). this case is considered by Goh + 0 T ( Qx(t), x(t) + (1966) and+ Dmitruk Further, if the cost contains x ˙ =“standard” Ax Bu,Legendre and(2008). thisterm case is considered by Goh + 0 ( Qx(t), x(t) + the (Ru, u) ,, then the (1966) and+Dmitruk Further, if the costfrom contains the “standard” Legendre term (Ru, u) then from the P (t)x(t), u(t) ) dt, (1) + 0 ( Qx(t),+ x(t) + (1966) andΩDmitruk (2008). Further, ifget the cost contains condition ≥ 0 on K one can easily the “classical” + P (t)x(t), u(t) ) dt, (1) the “standard” Legendre term (Ru, u) , then from the 0 condition Ω ≥ 0 on K one can easily get the “classical” + P (t)x(t), u(t) ) dt, (1) the “standard” Legendre term (Ru, u) , But then from the Legendre condition R(t) ≥ 0 for a.a. t. in our case condition Ω ≥ 0 on K one can easily get the “classical” + P (t)x(t), u(t) )function, dt, (1) Legendre condition R(t) ≥ can 0 foreasily a.a. t. But in“classical” our case where condition Ω ≥ 0 on K one get the where u(·) u(·) is is measurable measurable bounded bounded r−dimensional r−dimensional function, R(t) ≡ 0. Legendre R(t) ≥ 0 for a.a. t. But in our case R(t) ≡ 0. condition and x(·) is Lipschitz n−dimensional function T where u(·) measurable bounded r−dimensional function, Legendre and x(·) is is Lipschitz n−dimensional function on on [0, [0, T ]] that that R(t) ≡ 0. condition R(t) ≥ 0 for a.a. t. But in our case where u(·) is measurable bounded r−dimensional function, satisfy following linear equation of[0,Volterra Nevertheless, like and x(·)the is Lipschitz T ] that R(t) ≡ 0. satisfy followingn−dimensional linear integral integralfunction equationon like in in ODE ODE case, case, there there are are possible possible the the and x(·)the is Lipschitz n−dimensional function onof [0,Volterra T ] that Nevertheless, type further conditions, which are abscent in the general satisfy the following linear integral equation of Volterra Nevertheless, like in ODE case, there are possiblecase. the type further conditions, which are abscent in the general case. satisfy like in ODE there are general possiblecase. the type the following linear integral equation of Volterra Nevertheless, further conditions, which arecase, abscent in the Introduce the space type further conditions, which are abscent in the general case. Introduce the space t Introduce the space t t (A(t, τ )x(τ ) + B(t, τ )u(τ )) dτ, (2) Introduce the space x(t) = x + 0 x(t) = x0 + 0 t (A(t, τ )x(τ ) + B(t, τ )u(τ )) dτ, (2) W = Lipnn [0, T ] × Lrr∞ [0, T ] x(t) = x0 + 0 (A(t, τ )x(τ ) + B(t, τ )u(τ )) dτ, (2) W = Lipn [0, T ] × Lr∞ [0, T ] W = Lipn [0, T ] × L∞ [0, nT ] x(t) = is xa0quadratic + 0 (A(t, τ )x(τ ) +variables. B(t, τ )u(τ )) dτ, (2) and g(·, ·) form of 2n with elements W w = Lip (x, u), [0, nT ]is the space of [0, Twhere ] × Lr∞Lip and g(·, ·) is a quadratic form of 2n variables. 0 is the space of with elements w = (x, u), where Lip and g(·,matrix ·) is a quadratic form P of(t), 2n A(t, variables. n x(t), Lipschitz continuous n-vector functions and Lrr∞ of is All the functions Q(t), τ ) and B(t, τ ) are is the space with elements w = (x, u), where Lip andthe g(·,matrix ·) is a quadratic form P of(t), 2n A(t, variables. n x(t), Lipschitz continuous n-vector functions and L is All functions Q(t), τ ) and B(t, τ ) are the ∞ of isfunctions the space withspace elements w = (x, u), where Lip x(t), r of measurable bounded r-vector u(t). sufficiently smooth and uniformly bounded in t, τ . Lipschitz continuous n-vector functions and L All the matrix functions Q(t), P (t), A(t, τ ) and B(t, τ ) are ∞ r is the space of measurable bounded r-vector functions u(t). sufficiently smooth and uniformly bounded in t, τ . Lipschitz n-vector functions and Lu(t). All the matrix functions Q(t), P (t),bounded A(t, τ ) and ∞ is the space continuous of measurable bounded r-vectorx(t), functions sufficiently smooth and uniformly in t,B(t, τ . τ ) are We are the question sign-definiteness bounded r-vector functions u(t). sufficiently smoothin bounded in t, τ . of We are interested interested inand theuniformly question of of sign-definiteness of Ω Ω the space of measurable 2. on the of equation (2) satisfying the We are set interested in theto sign-definiteness Ω 2. THE THE CASE CASE OF OF ODE ODE on the set of solutions solutions toquestion equationof (2) satisfying also alsoof the We are interested in the question of sign-definiteness of Ω 2. THE CASE OF ODE following boundary conditions: on the set of solutions to equation (2) satisfying also the following conditions: 2. case THEwhen CASE on the setboundary of solutions to equation (2) satisfying also the Consider first the A OF and ODE B do not depend on following boundary conditions: Consider first the case when A and B do not depend on following boundary conditions: the first argument, i.e. A = A(τ ), B =do B(τ ). Consider first the case when A and not depend the on µ0 x(0) + µT x(T ) ≤ 0, the first argument, i.e. when A = A(τ ), BB B(τ ). Hence, Hence, Consider first the(2) case A to and B= do not depend the on integral equation is reduced the ODE µ0 x(0) + µT x(T ) ≤ 0, the first argument, i.e. A = A(τ ), B = B(τ ). Hence, the integral (2)i.e. is reduced (3) µ0 x(0) + µT x(T ) ≤ 0, the first equation argument, A = A(τto ), the B =ODE B(τ ). Hence, the (3) integral equation (2) is reduced to the ODE x(0) + µηη000x(0) x(T ))) = ≤ 0, 0, (3) integral equation (2) is reduced to the ODE x(0) + + µηηTTT x(T x(T = 0, (3) η0 x(0) + ηT x(T ) = 0, x x(0) + ηTofx(T ) = 0, x˙˙ = = Ax Ax + + Bu, Bu, x(0) x(0) = =x x00 .. where µ, η are someη0matrices corresponding dimensions. where µ, η are some matrices of corresponding dimensions. x ˙ = Ax + Bu, x(0) = x0 . where µ, η are some matrices of corresponding dimensions. The set L of all x˙the satisfying = pairs Ax + (x, Bu,u) x(0) = x0(2) . is The set L of all the pairs (x, u) satisfying (2) is a a subspace subspace where µ, η are some matrices of corresponding dimensions. This work is financially supported in part by the Russian Foundain W , and the set K of all the pairs (x, u) satisfying (3) The set L of all the pairs (x, u) satisfying (2) is a subspace This work is financially supported in part by the Russian Foundain W , and the set K of all the pairs (x, u) satisfying (3) is is The set L of all theK pairs (x, u)pairs satisfying (2) is a subspace a finite-faced cone in L. tion forwork BasicisResearch, projects no. in 16-01-00585 and 18-31-00091. in W , and the set of all the (x, u) satisfying is This financially supported part by the Russian Foundaainfinite-faced cone in of L.all the pairs (x, u) satisfying (3) tion for Basic Research, projects no. 16-01-00585 and 18-31-00091. W , and the set K (3) is This financiallyprojects supported part by theand Russian Foundaa finite-faced cone in L. tion forwork BasicisResearch, no. in 16-01-00585 18-31-00091. a finite-faced cone in L. tion for Basic Research, projects no. 16-01-00585 and 18-31-00091. 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2018 IFAC 424 Copyright 2018 IFAC 424 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 424 10.1016/j.ifacol.2018.11.421 Copyright © 2018 IFAC 424
IFAC CAO 2018 Anastasia Samylovskaya / IFAC PapersOnLine 51-32 (2018) 424–427 Yekaterinburg, Russia, October 15-19, 2018
As is known, in this case one can make a change of variables, in terms of which the quadratic form will contain the Legendre term with respect to some new control. First, introduce new state variables ξ, y satisfying system
y˙ = u,
y(0) = 0,
Introduce the matrix C(t) = B (t, t) and a new state variable ξ(t) = x(t) − C(t)y(t), where we still have y˙ = u, y(0) = 0,
ξ = x − Bu.
x(0) = ξ(0) = x0 = ξ0 .
The passage x −→ (ξ, y) is called Goh transform. Thus, we get
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Then we substitute x = ξ + Cy to (2) and obtain the expression
ξ˙ = Aξ + AB − B˙ y. Rewriting Ω in new variables (ξ, y, u), we get the following expression:
ξ(t) = ξ0 +
t
(A(t, τ )ξ(τ ) + D(t, τ )y(τ )) dτ,
(5)
0
where Ω(ξ, y, u) = g1 (ξ(0), ξ(T ), y(T )) + T + ( Q1 ξ, ξ + P1 ξ, y + 0
+ Gy, u + Ry, y) dt,
(4)
where Q1 , P1 , R are some matrices, and g1 is a quadratic form of 2n + r arguments. Represent G in form of G = S + V , where S is symmetric, and V is skew-symmetric matrices. Using the equality ˙ y) + 2(Sy, y), (Sy, y)˙ = (Sy, ˙ we obtain T 0
D(t, τ ) = A(t, τ )C(t) − Bτ (t, τ ). As we can see, (5) is quite similar to (2). Thus, from the initial pair of variables (x(t), u(t)) we come to a triple (ξ(t), y(t), u(t)). Rewriting Ω in these new terms, we receive the following expression: Ω(ξ, y, u) = g1 (ξ(0), ξ(T ), y(T )) + T + ( Qξ, ξ + P1 ξ, y + 0
+ P1 ξ, u + R1 y, y + Gy, u) dt.
1 1 T ˙ (Sy, y)dt ˙ = Sy, y − (Sy, y)dt, 2 2 0 T
and all these summends ”fit in” well to the form of (4) without u. So, the only part of G which remains in (4) is the skewsymmetric part V . A key fact (see Goh (1966) and Dmitruk and Shishov (2010)) is that the nonnegativity of Ω on K implies that V (t) ≡ 0 on [0, T ]. So, we now can take y as a new control belonging to Lr2 [0, T ] instead of being Lipschitz-continuous, and one can show that this extension will not affect the nonnegativity of Ω. Therefore, Ω ≥ 0 on K implies the Legendre condition R(t) ≥ 0 on [0, T ], which is called, in our situation, the Legendre-Goh condition. Note also that instead of y(T ) one should take a parameter h ∈ Rr , for which relations (3) fulfill after substitution x(t) = ξ(t) + B(t)h. Thus, we have a situation that differs from the “classical” one only by the presence of parameter h. The signdefiniteness of such a degenerated functional involving the parameter h is studied in Dmitruk and Shishov (2010). 3. THE GENERAL CASE
Our nearest goal is to exclude the control u from Ω. Consider the term P1 ξ, u separately and obtain
T
0
Clearly, the first two terms ”fit in” well to the form of (4). Let us then consider the last one. In view of (5), ξ˙ = A(t, t)ξ(t) + D(t, t)y(t) + t + At (t, τ )ξ(τ ) + Dt (t, τ )y(τ )dτ. 0
Introduce one more state variable t At (t, τ )ξ(τ ) + Dt (t, τ )y(τ )dτ. z(t) = 0
Notice that z(0) = 0. Then ˙ = A(t)ξ(t) + D(t)y(t) + z(t), ξ(t) and so we obtain
0
Let us try to act in a way similar to the one used in the case of ODE. 425
P1 (t)ξ(t), y(t) ˙ = P1 (T )ξ(T ), y(T ) − T ˙ − P˙1 (t)ξ(t), y(t) + P1 ξ(t), y(t) dt.
0
T
˙ P1 ξ(t), y(t) dt =
0
T
P1 z(t), y(t) +
+ P2 Aξ(t), y(t) + P3 Dy(t), y(t) dt.
IFAC CAO 2018 426 Anastasia Samylovskaya / IFAC PapersOnLine 51-32 (2018) 424–427 Yekaterinburg, Russia, October 15-19, 2018
Thus, instead of one phase variable x(t) we now have a triple (ξ(t), y(t), z(t)). Herewith the cost will take the form Ω(ξ, y, z, u) = g2 (ξ(0), ξ(T ), y(T )) + T + Q(t)ξ(t), ξ(t) + P3 (t)ξ(t), y(t) + 0
+ P1 (t)y(t), z(t) + R1 (t)y(t), y(t) +
is dense in K.
+ G(t)y(t), u(t) dt. Notice that ξ and z satisfy integral equations of the same type and introduce ρ = (ξ, z). Hence, we have
ρ(t) = ρ(0) +
t 0
Γ(t, τ )ρ(τ ) + Λ(t, τ )y(τ ) dτ,
In the proof, we use the following property (Dmitruk and Shishov (2010)). Lemma 3.1. (Density Lemma). Let L be a locally convex space, K be a finite-faced cone in L, and L ⊂ L be a linear manifold dense in L. Then = K ∩ L K
(6)
where Γ and Λ are some new matrices, the specific form of which is not important for our purposes. Rewrite Ω in terms of (ρ, y, u):
Since we got rid of u in Ω, hence Theorem 3.1 yields the following Corollary 3.1. (Goh-Legendre condition) If Ω ≥ 0 on [0, T ], then R(t) ≥ 0 on [0, T ]. Finally, by the analogy with Section 2, one can show that the space y ∈ C can be extended to y ∈ L2 with the change of the terminal value y(T ) by an arbitrary vector h ∈ Rr . Then we have Ω(ρ, y, h) = g˜ (ρ(0), ρ(T ), h) + T Q(t)ρ(t), ρ(t) + P (t)ρ(t), y(t) + + 0
Ω(ρ, y, u) = g˜ (ρ(0), ρ(T ), y(T )) + T ˜ + Q(t)ρ(t), ρ(t) + P˜ (t)ρ(t), y(t) + 0 + R(t)y(t), y(t) + G(t)y(t), u(t) dt,
(7)
+ R(t)y(t), y(t) dt.
and we obtain one more condition concerning the parameter h: Corollary 3.2. For any pair (ρ, y) satisfying (6) the following should hold: inf Ω(ρ, y, h) ≥ 0,
By the analogy with the above, the matrix G(t) can be reduced to a skew-symmetric matrix V (t), so
h
where inf is taken over all h ∈ Rr satisfying the constraints Ω(ρ, y, u) = g˜ (ρ(0), ρ(T ), y(T )) + T + Q(t)ρ(t), ρ(t) + P (t)ρ(t), y(t) + 0
+ R(t)y(t), y(t) + V (t)y(t), u(t) dt.
Introduce the space
with elements
(8)
= C n × C r × Lr W ∞
σ = (ρ, y, u) and rewrite (3) in terms of our current variables. Then we obtain
M0 ρ(0) + MT ρ(T ) + My y(T ) ≤ 0, N0 ρ(0) + NT ρ(T ) + Ny y(T ) = 0,
(9)
where M, N are some matrices of corresponding dimensions. The set L of all the pairs (ρ, y) satisfying (6) is of all the pairs (ρ, y) a subspace in W , and the set K satisfying (9) is a finite-faced cone in W . Then the following statement takes place. then V (t) ≡ 0. Theorem 3.1. If Ω ≥ 0 on K,
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M0 ρ(0) + MT ρ(T ) + My h ≤ 0, N0 ρ(0) + NT ρ(T ) + Ny h = 0. 4. CONCLUSION
Here we studied a question of sign-definiteness of a quadratic cost subject to the system of linear integral equations which has no Legendre term, i.e. is totally degenerate. Therefore, it cannot be studied by the clasical theory of quadratic forms. However, applying some generalizations of the known Goh transform, it is possible to reduce the given functional to a nondegenerate one, and hence, to obtain new necessary conditions of its nonnegativity. We reduced the original problem to a form which differs from the “classical” one only by parameter which defines endpoint value of a new control, by using Goh transformation (Goh (1966)). Thus, a new class of problems with reduced cost is investigated and corresponding necessary optimality conditions are obtained. ACKNOWLEDGEMENTS The author thanks Prof. Andrei V. Dmitruk for valuable remarks and discussions.
IFAC CAO 2018 Anastasia Samylovskaya / IFAC PapersOnLine 51-32 (2018) 424–427 Yekaterinburg, Russia, October 15-19, 2018
REFERENCES Dmitruk, A. (2008). Jacobi type conditions for singular extremals. Control and Cybernetics, 37(2), 285–306. Dmitruk, A. and Shishov, K. (2010). Analysis of a quadratic functional with a partly singular legendre condition. Moscow University Computational Mathematics and Cybernetics, 34(1), 16–25. Goh, B. (1966). Necessary conditions for singular extremals involving multiple control variable. SIAM J. Control, 4(2), 716–731.
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