Invariant stabilization of a certain class of functional differential equations
The system of functional differential equations \[ \dot x=A(\cdot)x+b(\cdot)u,\;\;u=s^{\top}(\cdot)x \] is considered with coefficients which are functionals of arbitrary nature, of which only the range of their variation is known. The first problem is to find \(b(\cdot)\) and \(s(\cdot)\) for which the system is globally asymptotically stable. Six classes of systems are considered for which this problem can be solved by using a quadratic Lyapunov function with constant coefficient-matrix. Also, the system \[ \dot x=A(\cdot)x+b_1(\cdot)u_1+b_2(\cdot)u_2+g(\cdot) \] is considered with the same restrictions on the right-hand side. The second problem is to determine controls \(u_1(\cdot)\) and \(u_2(\cdot)\) such that, for any initial point and for any \(g(\cdot)\), the output of the system \(\sigma=(c,x)\) would satisfy the invariance condition \(\dot \sigma+\varepsilon\sigma=0\) and boundedness condition \(\overline{\lim}_{t\to\infty}\|x(t)\|\leqslant\kappa\overline{\lim}_{t\to\infty}\|g(\cdot)\|\). In the paper, \(u_1\) and \(u_2\) are obtained provided the matrix \(A(\cdot)\) lies in either of the classes considered.
scientific article; zbMATH DE number 6109210