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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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