Quasistationary optimization in nonlinear systems using singularly perturbed equations (Q1180919)

Consider a dynamically controlled plant described by (1) \(dx/dt=H(x,u,a)\) where \(a\) is a perturbation vector. The set of equilibria \((x,u)\) of (1) is defined by (2) \(H(x,u,a)=0\). The problem under consideration is to minimize an objective function \(F(x,u)\) subject to the constraint (2). To this end it is assumed for each \(u=\text{const}\in R^ n\) and for each \(a\) from some bounded set \(\Omega_ 0\), (1) has a globally asymptotically stable equilibrium \(x=f(u,a)\). The authors consider two classes of plants (1): (i) \(dx/dt=AH(x,u)+b\), where \(A\) and \(b\) are perturbations, (ii) \(dx/dt=f(x)+Ku\) where estimates for \(f\) and \(K\) are know. They look for control laws of the type \(du/dt=\beta g(x,u,\hat a,\mu)\), where \(\hat a\) is an estimate of \(a\), \(\beta=\mu\) or \(\beta=\mu^{-1}\), where \(\mu\) is a small parameter under some additional conditions they solve the considered problem.