## Some characterizations of optimal trajectories in control theory.(English)Zbl 0744.49011

The paper provides a number of results related to optimal trajectory characterizations for the classical Mayer problem in optimal control theory. For instance, it is shown that for smooth control systems the value function $$V(t,x(t))$$ is continuously differentiable along an optimal state trajectory provided $$V$$ is differentiable at the initial point $$(t_ 0,x(t_ 0))$$. Then the upper semicontinuity of the optimal feedback map is deduced. In particular, it is shown that whenever the feedback map is single-valued, it is continuous. The problem of optimal design is adressed, obtaining sufficient conditions for optimality. Moreover, it is shown that the optimal control problem may be reduced to a viability one. Finally, the case involving endpoint constraints is treated via penalization techniques and it is shown that the value function of such a problem may be approximated by the value function of problems with free endpoints.

### MSC:

 49K15 Optimality conditions for problems involving ordinary differential equations 49L99 Hamilton-Jacobi theories 93B50 Synthesis problems 93C15 Control/observation systems governed by ordinary differential equations
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