On the existence of optimal solutions for infinite horizon optimal control problems: Nonconvex and multicriteria problems. (English) Zbl 0619.49002

Carlson, Dean A.

Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 77, 21-27 (1984).

This note refers to the optimal control problem \(\dot x=f(t,x,u)\), \(x(0)=x_ 0\), with constraints on the control u(t)\(\in U(t,x)\) and on the state (t,x(t))\(\in A\). The cost functional is of the infinite horizon type \(\int^{\infty}_{0}g(t,x,u)dt\), which is supposed to be convergent for all admissible solutions. In addition, standard assumptions are made.
First, for the non-convex case, the related ”relaxed” problem is defined and Cesari-type ”conditions Q” are introduced. This way two existence theorems are given. For the case that f(.,.,.) and g(.,.,.) are linear in the state x, a stronger theorem is given. Finally, a result for multicriteria optimality, which is then taken in the sense of Pareto optimality. For the details of the proofs, the reader is referred to the author’s dissertation and a future paper.

Reviewer: E.Roxin

MSC:

49J15	Existence theories for optimal control problems involving ordinary differential equations
58E17	Multiobjective variational problems, Pareto optimality, applications to economics, etc.
90C31	Sensitivity, stability, parametric optimization
34H05	Control problems involving ordinary differential equations
93C10	Nonlinear systems in control theory
93C15	Control/observation systems governed by ordinary differential equations

Keywords:

infinite horizon; ”relaxed” problem; Cesari-type ”conditions Q”; multicriteria optimality; Pareto optimality