Structural properties of minimax controllers for a class of differential games arising in nonlinear \(H^ \infty\) control
The authors deal with a minimax control problem described by the equation \[ \dot x= f(t,x,u,w),\quad x(0)= x_ 0,\quad t\in [0,T],\tag{1} \] where \(x\in \mathbb{R}^ n\) is a state vector, \(u\in U\subset\mathbb{R}^ m\) is a control. It is assumed that \[ (x_ 0,w(\cdot))\in \Omega\subset\mathbb{R}^ n\times {\mathcal L}_ 2([0,T],\mathbb{R}^ I), \] i.e., both the initial state \(x_ 0\) and the disturbance \(w(\cdot)\) are not known. The controller choosing \(u\) has access to only a noise-corrupted output \(y\) in \(\mathbb{R}^ p\), given by \(y(t)= h(t,x(t))+ v(t)\), where \(v(\cdot)\in {\mathcal V}\subset {\mathcal L}_ 2([0,T],\mathbb{R}^ p)\) is part of general disturbance. Here \(\Omega\) and \(\mathcal V\) are some specified sets. An admissible control strategy is defined as a nonanticipatory (casual) map \(y(\cdot)\mapsto \mu(y(\cdot))\) such that the differential equation (1) with \(u(\cdot)= \mu(y(\cdot))\) has a solution for every \((x_ 0,w(\cdot))\in \Omega\) and every \(v(\cdot)\in {\mathcal V}\). The cost functional \(J\) is given by \[ J= q_ T(x(T))+ \int^ T_ 0 [q(t,x(t),u(t),w(t))- r(t,v(t))]dt- q_ 0(x_ 0). \] The objective is to find a minimax strategy for the controller that minimizes the supremum of \(J\) over all possible disturbances. This problem is considered under certain assumptions. In particular, it is assumed that there exists a solution \(V\) of class \(C^ 1\) of a Cauchy problem for Isaacs' equation \[ -{\partial V\over \partial t}= \inf_ u \sup_ w \left\{{\partial V\over \partial x} f(t,x,u,w)+ q(t,x,u,w)\right\},\quad V(T,x)= q_ T(x). \] It is also assumed the uniqueness of solutions of some extremal problems. Examples satisfying these requirements are not considered.
scientific article; zbMATH DE number 550117