A simple `finite approximations' proofs of the Pontryagin maximum principle under reduced differentiability hypotheses

The authors consider the optimal control problem with terminal constraints cases J(x,u)=psi^0(x(S),x(T))+int^T_ SL(t,x(t),u(t))dtto min, dot x(t)=f(t,x(t),u(t))quad {rm for} {rm almost} {rm all} tin [S,T], u(t)in U(t)quad {rm for} {rm almost} {rm all} tin [S,T], psi^j(x(S),x(T))leq0quad {rm for} j=1,dots, r_1, phi^j(x(S),x(T))=0 quad {rm for} j=1,dots, r_2,endcasestag1 in which the data are the interval [S,T], the functions f(cdot,cdot,cdot)colon[S,T]times {bf R}^ntimes{bf R}^mto{bf R}^n, L(cdot,cdot, cdot)colon[S,T]times{bf R}^ntimes{bf R}^mtobf R, psi^j(cdot,cdot)colon{bf R}^ntimes{bf R}^ntobf R (j=0,dots,r_1), and phi^j(cdot,cdot)colon{bf R}^ntimes{bf R}^nto bf R (j=1,dots,r_2), and the multivalued mapping U(t)colon [S,T]rightsquigarrow {bf R}^m. It is assumed that U(cdot) is measurable, that for any pair (x,u)in{bf R}^ntimes {bf R}^m the functions f(cdot,x,u) and L(cdot,x,u) are measurable, and that for any tin[S,T] the functions f(t,cdot,cdot) and L(t,cdot,cdot) are continuous. The controls u(cdot)colon[S,T]to{bf R}^m are assumed to be measurable, and the functions x(cdot)colon [S,T]to{bf R}^n are absolutely continuous. The pair (u(cdot),x(cdot)) satisfying the constraint dot x(cdot)=f(cdot,x(cdot),u(cdot)) is called a process, and the function x(cdot) is called the phase trajectory corresponding to the control u(cdot). A process satisfying all the constraints of problem (1) is called an admissible process. par Assume the existence of an admissible process overline u(cdot), overline x(cdot) and a number epsilon >0 such that the following conditions are satisfied: all functions psi^j and phi^j are Fréchet differentiable at the point (overline x(S),overline x(T)); all functions phi^j are continuous in some neighborhood of (overline x(S),overline x(T)); whatever the point x_0in overline x(S)+epsilon B, where B is the unit ball with center at the origin of the space {bf R}^n, at least one phase trajectory x(cdot) corresponds to any control u(cdot) such that x(S)=x_0 and |x(cdot)-overline x(cdot)|_Cleqepsilon; for almost every tin [S,T] the functions f(t,cdot,overline u(t)) and L(t,cdot,overline u(t)) are Fréchet differentiable at the point overline x(t); there exists a function k(cdot)in L_1 such that for almost every tin [S,T] and for any xin overline x(t)+epsilon B, |f(t,x,overline u(t))-f(t,overline x(t),overline u(t))|leq k(t) |x-overline x(t)|, |L(t,x,overline u(t))-L(t,overline x(t),overline u(t))|leq k(t) |x-overline x(t)|. The authors prove that if under these conditions the admissible process overline u(cdot), overline x(cdot) yields a strong local minimum in problem (1), then it satisfies the conditions of the Pontryagin maximum principle. In addition, they prove that if the admissible process overline u(cdot), overline x(cdot) yields a Pontryagin local minimum in problem (1) when, in addition to those conditions, there exists a function c_f(cdot)in L_1 such that for almost every tin [S,T] for any xin overline x(t)+epsilon B and any uin U(t) we have |f(t,x,u)|leq c_f(t),quad |L(t,x,u)|leq c_f(t), then the process overline u(cdot), overline x(cdot) also satisfies the conditions of the Pontryagin maximum principle. par The authors note that the optimality conditions they proved are in themselves not new and are similar to the conditions proved by H. J. Sussmann (under somewhat different assumptions) [see, for example, in {it Nonlinear control in the year 2000, Vol. 2 (Paris)}, 487--526, Lecture Notes in Control and Inform. Sci., 259, Springer, London, 2001; [msn] MR1806192 (2002e:49040) [/msn]; in {it Mathematical control theory}, 140--198, Springer, New York, 1999; [msn] MR1661472 (99k:49051) [/msn]]. What is new is the method for deriving the optimality conditions. It consists in a finite-dimensional approximation of problem (1), the application of the finite-dimensional Lagrange multiplier rule, and the subsequent passage to the limit. The rule of Lagrange multipliers used in this paper is similar to the one proved by H. Halkin [SIAM J. Control {bf 12} (1974), 229--236; [msn] MR0406524 (53 #10311) [/msn]]. The method of finite-dimensional approximation used goes back conceptually to the well-known Tikhomirov scheme for proving the Pontryagin maximum principle under standard differentiability assumptions [see V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, {it Optimal control} (Russian), "Nauka", Moscow, 1979; [msn] MR0566022 (81g:49001) [/msn]]. The authors show (and this is, as they note, the main achievement of the paper) that a modification of Tikhomirov's scheme using, in particular, the above-mentioned fairly refined version of the finite-dimensional multiplier rule enables one to obtain the Pontryagin maximum principle under the above-mentioned general assumptions about the data of the optimal control problem.