Using transformation methods to describe discontinuous trajectories in problems with phase constraints (Q5951178)

The author considers the minimization problem for the lower semicontinuous functional \[ J[x]=\Phi(x(T)) \tag{1} \] on the trajectories of the differential inclusion \[ \dot{x}\in F(t,x) \tag{2} \] satisfying the end- and interior-point phase constraints \[ x(0)\in C_{0},\quad x(T)\in C_{T}, \tag{3} \] \[ x(t)\in A\quad \forall t\in [0,T]. \tag{4} \] Here, \(C_{0},C_{T},\) and \(A\) are given closed subsets of \(\mathbb{R}^{n}.\) The multimapping \(F\) takes points \((t,x)\) to unbounded, possibly nonconvex subsets of \(\mathbb{R}^{n}.\) Just as in the case of optimal control problems with linear unbounded controls for problem (1)-(4) with an unbounded differential inclusion typically, there is no trajectory at which the minimum of \(J\) is attained, and a minimizing sequence \(\{x_{n}\}\) converges to a discontinuous function. In this paper, discontinuous trajectories of the unbounded differential inclusion (2) are introduced, and approximation theorems for problem (1)--(4) are proved.