Tonelli Full-Regularity in the Calculus of Variations and Optimal Control

Transcription

1 Tonelli Full-Regularity in the Calculus of Variations and Optimal Control Delfim F. M. Torres Department of Mathematics University of Aveiro Aveiro, Portugal

2 Lagrange Problem of Optimal Control with unrestricted controls b a L (t, x (t),u(t)) dt min ẋ (t) =ϕ (t, x (t),u(t)) (P ) x (a) =A, x (b) =B u ( ) L 1 ([a, b]; IR r ), x ( ) W 1,1 ([a, b]; IR n ) L :[a, b] IR n IR r IR ϕ :[a, b] IR n IR r IR n L(,, ), ϕ(,, ) C L(,,u), ϕ(,,u) C 1 May 3 4, Coimbra 1

3 Necessary Optimality Condition for (P) Pontryagin Maximum Principle (PMP) Definition. The quadruple (x ( ),u( ),ψ 0,ψ( )), ψ 0 0, ψ ( ) W 1,1,(ψ 0,ψ( )) 0,isa Pontryagin extremal if it satisfies: the Hamiltonian system ẋ = H ψ, The maximality condition ψ = H x H (t, x (t),u(t),ψ 0,ψ(t)) = sup u IR r H (t, x (t),u,ψ 0,ψ(t)) with the Hamiltonian H = ψ 0 L (t, x, u)+ ψ, ϕ (t, x, u). PMP. Under (H1) or (H2) or... all minimizers are extremals. (H1) L(,x, ), ϕ(,x, ) C, L(t,,u), ϕ(t,,u) C 1 ; u( ) L (H2) L(,x, ), ϕ(,x, ) Borel measurable; u ( ) L 1 ; c>0andk L x c L + k, ϕ i x c ϕ i + k May 3 4, Coimbra 2

4 You shall not be naive Any theory of necessary optimality conditions is naive until the existence of optimal solutions is clarified. Perron Paradox Theorem. A necessary condition for N to be the largest positive integer is that N =1. Proof. If N 1 then N 2 >N.SoN is not the largest integer, contrary to the hypothesis. Thus, N = 1. There is nothing wrong with this statement = Necessary conditions in optimization may be useless if we do not know that the solution we are talking about exists one may derive a wrong conclusion from a correct necessary condition. May 3 4, Coimbra 3

5 Tonelli Existence Theorem for (P) Theorem. Problem (P) has an absolute minimum in the space u( ) L 1, provided that there exist at least one admissible pair, and the following conditions are satisfied for all (t, x, u). Coercivity: there exists a function θ :IR + 0 below, such that θ (r) lim =+, r + r IR, bounded L (t, x, u) θ ( ϕ(t, x, u) ), lim ϕ(t, x, u) =+. u + Convexity: L (t, x, u) andϕ (t, x, u) are convex w.r.t. u. May 3 4, Coimbra 4

6 The standard method to solve (P) 1. Prove that a solution to the problem exists. 2. Assure that some regularity conditions hold, implying the applicability of necessary optimality conditions. 3. Apply the necessary conditions to identify the extremals (the candidates). Further elimination, if necessary, identifies the minimizer(s) of the problem. Question Is Step 2 really necessary? In particular: does the Pontryagin Maximum Principle hold under just the hypotheses of Tonelli s Existence Theorem? May 3 4, Coimbra 5

7 Tonelli Regularity Theorem. Assume that the Tonelli Existence Hypotheses are satisfied. Take any minimizer ( x( ), ũ( )) of (P ). Then there exists a closed subset Ω [a, b] of zero measure with the following property: for any τ [a, b] \ Ω, ũ(τ) is essentially bounded on a relative neighborhood of τ. Corollary. The Pontryagin Maximum Principle is valid on relatively open subset of [a, b], of full measure. Previous question remain Is it really possible to the Ω set to be nonempty? Is it possible that such ( x( ), ũ( )) fail to be an extremal? May 3 4, Coimbra 6

8 La vita è bella Answer: Yes. Bad behaviour do exist. Ω may be nonempty. Even for polynomial Lagrangians and linear dynamics, minimizers predicted by existence theory may fail to be Pontryagin extremals. What can be done? How to exclude the possibility of bad behaviour? How to obtain full-regularity (Ω = )? Postulate conditions beyond those of the Existence Theorem, assuring that all minimizing controls are bounded. Validity of classical necessary optimality conditions Lipschitzian Regularity of Minimizing Trajectories Possibility of discretization and numerical procedures May 3 4, Coimbra 7

9 Tonelli Full-Regularity Classes of well-behaved problems Theorem. (L. Tonelli& C. B. Morrey) For the basic problem of the Calculus of Variations (CV), b a L (t, x (t),u(t)) dt min, ẋ (t) =u (t), suppose that for certain constants c > 0 and k one has L x + L u c L + k. (1) Then any solution ũ( ) is essentially bounded. F. H. Clarke & R. B. Vinter proved that: The classical Tonelli-Morrey conditions (1) can be generalized: L x c L + k. (2) Tonelli-Morrey type conditions (2) work universally in the CV. May 3 4, Coimbra 8

10 Full-Regularity in Optimal Control Theorem. (A. V. Sarychev & D. F. M. Torres) For the Lagrange Problem of Optimal Control (P) with control affine dynamics, ϕ = f(t, x)+g(t, x) u,ifg(t, x) has complete rank r for all t and x; the coercivity condition holds; and γ>0, β<2, η, and µ max {β 2, 2}, such that ( L t + L x i + Lϕ t L t ϕ + Lϕ x i L x i ϕ ) u µ γl β + η, (3) then all the minimizers ũ ( ) of the problem, which are not abnormal extremal controls, are essentially bounded on [a, b]. Convexity is not required in the regularity theorem Conditions (3) are not of the type of Tonelli-Morrey. Results are possible for general nonlinear dynamics May 3 4, Coimbra 9

11 Main Result Tonelli-Morrey type conditions work universally in Optimal Control Theorem. Coercivity plus the growth conditions: there exist constants c > 0andk such that L L t c L + k, x c L + k, ϕ ϕ i t c ϕ + k, x c ϕ i + k (i =1,..., n); imply that all minimizers ũ( ) of(p ), which are not abnormal extremal controls, are essentially bounded on [a, b]. Corollary. Under the hypotheses of the Theorem, all minimizers of (P) are Pontryagin extremals. May 3 4, Coimbra 10

12 Problem (P). Our Approach to Full-Regularity b a L (t, x(t), u(t)) dt min, Problem (P τ [ w( )]). ẋ(t) =ϕ (t, x(t), u(t)) b J [t( ), z( ), v( )] = L (t(τ), z(τ), w(τ)) v(τ)dτ min a t (τ) =v(τ), v(τ) [0.5, 1.5] z (τ) =ϕ (t(τ), z(τ), w(τ)) v(τ) Proposition. If ( x(t), ũ(t)) is a minimizer of (P ), then the triple ( t(τ), z(τ), ṽ(τ) ) =(τ, x(τ), 1) furnishes a minimizer to (P τ [ũ( )]). Moreover, if ũ( ) is not an abnormal extremal control, then ṽ 1 is not an abnormal extremal control too. May 3 4, Coimbra 11

13 Proof of the Main Result We know that ( t(τ), z(τ), ṽ(τ) ) =(τ, x(τ), 1) is a normal extremal to problem (P τ [ũ( )]). From the maximality condition [ v L (τ, x(τ), ũ(τ)) + ψ t (τ)+ ψ ] z (τ) ϕ (τ, x(τ), ũ(τ)) v is maximized at v =1.Thisimpliesthat L (τ, x(τ), ũ(τ)) = ψ t (τ)+ ψ z (τ) ϕ (τ, x(τ), ũ(τ)). Let ψ t (τ) M and ψ z (τ) M on [a, b]. Dividing both sides of inequality by ϕ (τ, x(τ), ũ(τ)) and using the coercivity hypothesis one obtains θ ( ϕ (τ, x(τ), ũ(τ)) ) ϕ (τ, x(τ), ũ(τ)) M 1+ ϕ (τ, x(τ), ũ(τ)) ϕ (τ, x(τ), ũ(τ)) Coercivity yields the essential boundedness of ũ( ) on[a, b].. May 3 4, Coimbra 12

14 1 0 An Example ( u 2 1 (t)+u 2 2(t) ) ( e 2(x 1(t)+x 2 (t)) +1) x 1 (t) = u 2 1 (t)+u2 2 (t) x 2 (t) =u 2 (t)e x 1(t)+x 2 (t) dt min x 1 (0) = 0, x 1 (1) = 1, x 2 (0) = 1, x 2 (1) = 1. Dynamics is nonlinear both in the state and control variables All conditions of Tonelli s Existence Theorem are satisfied Previously known regularity conditions fail Our theorem allow us to conclude that all minimizing controls, which are not abnormal extremal controls, are bounded From our Corollary, all minimizing controls of the problem can be identified via the Pontryagin Maximum Principle May 3 4, Coimbra 13

15 Final Remarks Convexity is not required in the regularity theorem Results are valid for general problems of optimal control with nonlinear dynamics It provides Tonelli-Morrey type conditions which are easy to check in practice It is also possible to obtain new regularity conditions, which are not of the type of Tonelli-Morrey: Theorem. Assume the coercivity condition of Tonelli s existence theorem and that the Pontryagin maximum principle is applicable to (P τ [ w( )]). Then, all control minimizers ũ( ) of(p ), which are not abnormal extremal controls, are essentially bounded on [a, b]. May 3 4, Coimbra 14

16 My Contribution to the Call for Problems Open Question The question of how to establish Lipschitzian regularity for the abnormal minimizing trajectories seems to be completely open: For the problems of the calculus of variations studied by L. Tonelli, F. H. Clarke, R. B. Vinter, et. al., no abnormal extremals exist For the optimal control problems considered by A. V. Sarychev and D. F. M. Torres, abnormal extremals are, like here, put aside How to establish Lipschitzian regularity for the abnormal minimizing trajectories? May 3 4, Coimbra 15