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Existence theorems for optimal control and calculus of variations problems where the states can jump. (English) Zbl 0587.49004
This paper examines optimal control and calculus of variations problems where the states can be discontinuous. If we allow such trajectories to be feasible, we can no longer confine them to be absolutely continuous. A larger class of functions that is suitable is the space of bounded variations. Similarly, the controls go from being measurable functions to the derivatives of functions of bounded variation. The objective functional is then extended so that it can handle this larger class of states and controls and so that it matches up with the usual type of problem. It is also done in such a way that under certain conditions an optimal solution will exist. The major part of the paper is devoted to the proof of the existence theorem. Actually, two separate cases are treated, one where the costate constraint set P depends on the state x and one where it does not.
Many examples of problems that fit this framework can be found. For example in economics where the costates p represent prices one often has a restriction on the range of prices P. If the optimal costate p strikes the boundary of P a jump can occur in the optimal state x.

MSC:
49J15 Existence theories for optimal control problems involving ordinary differential equations
26A45 Functions of bounded variation, generalizations
49J45 Methods involving semicontinuity and convergence; relaxation
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