Analytical design of regulators for nonlinear control systems (Q1285466)

The problem of an analytic design of regulators for nonlinear control systems is considered. Let the motion of a guided object in \({\mathbb{R}}^n\) be described by the ordinary differential equations \[ \dot x = f(x) + B(x)u, \tag{1} \] where \( x\in {\mathbb{R}}^n\) is the phase vector, \(u \in {\mathbb{R}}^r\) is a guidance, the function \(f(x): {\mathbb{R}}^n \to {\mathbb{R}}^n\) and the matrix \(B(x): {\mathbb{R}}^n \to {\mathbb{R}}^{n\times r}\) are supposed to be smooth in \({\mathbb{R}}^n\). Additionally, the function \(f(x)\) can be written in a ``quasilinear'' form \[ f(x) = A(x)x, \tag{2} \] where \(A(x) \in {\mathbb{R}}^{n \times n}\) is the matrix with a smooth dependence on \(x\) . A quality criterion is written in a ``quasiquadratic'' form \[ L(u) = \int\limits_0^{+\infty}[x^TQ(x)x + u^TR(x)u]dt, \tag{3} \] where \(Q(x)\in {\mathbb{R}}^{n\times n}, R(x)\in {\mathbb{R}}^{r\times r}\) are symmetrical positive definite matrices. A class \(K\) of possible guidances \(v(x) : {\mathbb{R}}^n \to {\mathbb{R}}^r\) is defined. It is proved that there exist the a possible guidance \(u_0(x) \in K\) minimizing the functional (3) on the trajectories (1).