Constrained retarded nonlinear optimal control problems in Banach state space
The author considers the functional differential equation \[ x'(t)= f\bigl(x (\cdot), u(t),t\bigr) \tag{1} \] where \(f:C(0,1;X)\times U\times [0,1]\to X\) \((X\) a Banach space, \(C(0,1;X)\) the space of continuous \(X\)-valued functions furnished with the supremum norm, \(U\) a topological space). The function \(f\) is nonanticipating in the sense that, for a given \(t\), it only depends on past values of \(x(t)\). An optimal control problem is assembled as follows. Let \(Y,Z\) be Banach spaces, \(r:X\to Z\), \(g:C(0,1;X) \times U\times [0,1]\to Y\) another nonanticipating function, \(L\) a one-dimensional subspace of \(Z\), \(b\in Z\), \(\varphi: Z\to \mathbb{R}\) a linear functional, \(C\) a subset of \(Y\). The problem is to minimize the cost functional \(\varphi(r(x(1)))\) among the solutions of (1) satisfying an initial condition and \(r(x(1)) \in b+L\), \(g(x(\cdot), u(t),t)\in C\), where the first inclusion takes care of the target condition and the second of the state constraints, which may be of functional type. The apparent restriction that the cost functional should depend only on the final point of the trajectory is lifted (as in classical control theory) by introducing an additional ``cost coordinate''. The author shows necessary conditions of the type of Pontryagin's maximum principle including target conditions both of instantaneous type (in particular, point targets) and of functional type; as expected, if there is not much slack in the target conditions one must rely on controllability properties of the linearized system to guarantee nontriviality of the costate.
scientific article; zbMATH DE number 920738
scientific article