degenerate system of differential equations
Application of the Pontryagin maximum principle for linear degenerate optimal control problems
In this paper, an optimal control problem for a linear time-varying system with the quadratic cost functional depending on the control only is considered. The author assumes that such a linear system is degenerated in the sense that the matrix at the time-derivative is of constant rank less than the dimension of the state space. In order to solve the optimal control problem considered, a transformation that reduces the degenerated problem to a regular optimal control problem together with an additional boundary value problem is applied. Then the adjoint equation is introduced for the reduced optimal control problem. A necessary optimality condition is formulated in terms of a solution to the adjoint equation and the solvability of a degenerated boundary value problem. It is shown that the optimal control is unique if the boundary conditions satisfy certain compatibility conditions. These results justify the Pontryagin maximum principle for degenerated linear systems.
scientific article; zbMATH DE number 6062222