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

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