Optimal guaranteed control of linear systems under disturbances (Q1387314)

The author considers the following linear systems \[ \dot x(t)=A(t)x(t)-B(t)u(t)-D(t)w(t)+C(t)v(t), \tag{1} \] \(t\in [t^0,\theta], \;x(t^0)=x^0\), where \(A(t), B(t), C(t), D(t)\) are continuous matrices of dimensions \(n\times n\), \(n\times m\), \(n\times l\), \(n\times k\), respectively. Parameters \(u\) and \(w\) are controls, and \(v\) is a disturbance. It is assumed that \[ u(t)\in U(t), \tag{2} \] where \(t\mapsto U(t)\subset R_m\) is a continuous convex- and compact-valued mapping. The performance (cost) functional has the form \[ I=\gamma(x(\theta))+\int_\theta^{t^0} ( w_T (t)G(t)w(t)v_T (t)H(t)v(t)) dt, \tag{3} \] where \[ \;\gamma(x)=\min_{y\in M^0} (x-y)_T R^0 (x-y), \tag{4} \] \(G(t), H(t)\) are symmetric positive semidefinite continuous matrices, \(R^0\) is a positive definite matrix, \(M^0 \subset R_n\) is the convex closed target set. The aim of controls is minimizing the cost functional, the disturbance tries to maximize the cost functional. It is proven that the differential game has the value function, i.e., there exists the equilibrium in the classes of feedbacks: Borel measurable controls \(u(t,x)\) plus continuous controls \(w(t,x)\) and continuous disturbances \(v(t,x)\). In the paper, simple formulas for calculating the value function and optimal guaranteed feedbacks are obtained.