Optimal impulse control problem with restricted number of impulses

Abstract

The classical statement of the impulse control problems [73], [98], [122], [218], [121] presumes, as a rule, the energy type constraints to be imposed on the total intensity of control actions. However, if there are no restrictions that are imposed on the total number of impulses or/and on the impulse repetition rate, then the impulse sliding mode can appear as an optimal solutions (see [73], [122] and Examples 1.2 and 1.3). The realization of such modes needs the extremely high impulse repetition rate, which may be illegal in some technical systems. One of the available method with these constraints to be taken into account is to restate the optimal control problem as a problem of mathematical programming. In this way one can obtain the optimality conditions of a Khun-Tucker’s type and solve the problem somehow with the aid of numerical procedures. Meanwhile in the optimal control problems there are some more powerful tools, like the Pontriagin maximum principle [18], [59], [169] which is much more effective, than mathematical programming methods. However, to derive the maximum principle in DCS, one has to justify the convexity of the attainability set for the system state after one impulse application [3], [4], [206]. Generally it is rather a complicated problem which can be effectively solved only for linear systems.