Construction of robust control laws on the basis of the minimax approach (Q1874801)

The author deals with a control problem for a nonlinear dynamical system described by the equation \[ x'=Ax+Bu+F\varphi(x,t,u), \quad t\geq0,x(0)=x_0, \tag{1} \] where \(x\in\mathbb R^n\) is the state, \(u\in\mathbb R^m\) is the control, \(\varphi(x,t,u)\) is from the class \(\Xi\) of continuous functions satisfying the inequality \[ \int_0^T\Psi(x(t),u(t), \varphi(x(t),u(t),t))\, dt\geq0,\;\forall T\geq T \] with a given minimax quadratic form \(\Psi(x,u,w)\) and an arbitrary solution \(x(t)\) of (1) for the corresponding control law \(u(t)\). The problem is to construct robust state feedback control laws \[ u=-\Theta x \tag{2} \] such that the equilibrium \(x=0\) of the closed system (1), (2) is asymptotically stable for any \(\varphi(x,t,u)\) from the class \(\Xi\). It is shown that so-called locally minimax control laws in an appropriately constructed linear-quadratic differential game can be chosen as the desired robust control laws.