On linear-quadratic dissipative control processes with time-varying coefficients
In this paper, dissipative linear control systems of the form \[ \dot{x}(t) = A(t)x(t) + B(t)u(t),\qquad x(t) \in \mathbb{R}^n,\;u(t) \in \mathbb{R}^m, \] are studied, where the matrix-valued functions \(A(\cdot)\) and \(B(\cdot)\) are bounded and uniformly continuous. Dissipativity is described in terms of a supply rate, which is a function of the form \[ \mathcal{Q}(t,x,u) = \frac{1}{2}\left\{\langle x,G(t)x \rangle + 2\langle x,g(t)u \rangle + \langle u,R(t)u \rangle\right\} \] that measures the amount of energy the system receives from its environment. The system is dissipative if it cannot restore energy to the environment, which is described by the inequality \[ \int_{t_0}^{t_1} \mathcal{Q}(t,x(t),u(t))\mathrm{d}t \geq 0, \] where \(t_0<t_1\), \(u(\cdot)\) is a square-integrable control function, and \(x(\cdot)\) denotes the corresponding solution with \(x(t_0)=0\). One of the main questions investigated in the paper is whether a dissipative system admits a storage function, i.e., a continuous nonnegative function \(V:\mathbb{R}\times\mathbb{R}^n \rightarrow \mathbb{R}\) satisfying \(V(t,0) = 0\) and \[ \int_{t_0}^{t_1}\mathcal{Q}(t,x(t),u(t))\mathrm{d}t \geq V(t,x(t_1)) - V(t,x(t_0)) \] for every control \(u(\cdot)\) with corresponding solution \(x(\cdot)\) for an arbitrary initial value. Such \(V\) is considered as a measure for the energy stored inside the system. Using the Yakubovich frequency theorem, it was proved in [\textit{V. Yakubovich} et al., IEEE Trans.~Automat.~Control 52, No. 6, 1039--1047 (2007)] that a strongly dissipative system with a quadratic supply rate admits a quadratic storage function, provided that the coefficient matrices of both the system and the supply rate are periodic with a common period. The authors generalize this and related results from [loc. cit.] to the non-periodic non-autonomous case described above, using a more general version of the Yakubovich frequency theorem, proved in [\textit{R. Fabbri, R. Johnson} and \textit{C. Núñez}, Discrete Contin.~Dyn.~Syst.~9, No. 3, 677--704 (2003; Zbl 1028.37014)].
scientific article; zbMATH DE number 6146682