Nonlinear flexible control systems (Q1390692)

The author considers the following control problem. The equation of motion is: \[ dx/dt= f(x)+ Bu,\tag{1} \] where \(f\) is a nonlinear Lipschitz function defined for all \(x\in \mathbb{R}^n\), \(u\in\mathbb{R}^p\), \(B\) is a constant (real) \(n\times p\) matrix. It is known that if \(u\) is a continuous control then the above control problem has a unique solution \((x,u)\) on a time interval \(\Delta= [0,T]\). Let the output from the system (the reviewer would prefer to call it observation) be denoted by \[ y= \pi x(t),\quad t\in\Delta,\tag{2} \] where \(\pi\) is an \(n\times m\) matrix. Let \(L_2(m,\Delta)\) denote the Hilbert space of (Lebesgue) square integrable \(m\)-dimensional vectors defined on \(\Delta\). The system (1), (2) with appropriate Cauchy data is called a flexible control system if for any function \(g(.)\in L_2(m,\Delta)\) we have: \[ \inf_{u\in C(p,\Delta)}\|\pi x(.,u(.))- g\|=0. \] \(C(p,\Delta)\) is a Banach space of continuous functions on \(\Delta\), with a uniform norm. First, the author examines the linear case: \[ dx/dt= A_0x+ Bu,\tag{1\(^a\)} \] where \(A_0\) is an \(n\times n\) constant matrix. The following theorem is stated: This linear control system is flexible if and only if for a sufficiently small \(r\) the rank of the matrix \(\pi(I-rA_0)^{-1}B\) is equal to \(m\), that is, to the number of columns of the matrix \(\pi\). Note that \(I\) denotes the identity matrix. (The author uses the symbol \(E\).) It is clear that flexibility of the controlled system does not depend on the value of \(x_0\). However, this is not the case if the system is nonlinear. The author offers an example showing that for some choices of one initial component of his nonlinear system it is flexible, irrespective of the choices of other components, but that in general the property of flexibility depends on choices of initial conditions. Considering the general nonlinear case \(f(x)\), let \(A\) be an arbitrary \(n\times n\) matrix, let \(M\) denote the minimal (with respect to inclusion) subspace of \(\mathbb{R}^n\) which contains all vectors that can be represented in the form \(f(x)- f(0)- Ax\), \(x\in \mathbb{R}^n\). The pair \(\{x, Ax\}\) is said to linearize the vector field \(f(x)\). The following definition is offered: Let \(A\) be any \(n\times n\) matrix. The subspace \(M\subset \mathbb{R}^n\) is said to be subordinated to the subspace \(B\mathbb{R}^p\) by the use of the matrix \(A\) if for any sufficiently small \(r\geq 0\) and for some \(\alpha> 0\), the following inclusion is true: \(\alpha\pi(I- rA_0)^{-1} BS_p\supset \pi(I- rA_0)^{-1} LS_n\), where \(L\) is a matrix of orthogonal projection from \(\mathbb{R}^n\) into \(M\), and \(S_q= \{y\in \mathbb{R}^q:| y|\leq 1\}\). Comment: There is a fairly good likelihood of the nonexistence of even a single matrix \(A\) such that \(M(A)\) is subordinated by the use of the matrix \(A\) to \(B\mathbb{R}^p\). The author illustrates these ideas on a very simple example. The next important theorem states that if there exists a matrix \(A_0\) such that the linear system \(dz/dt= A_0z+ Bu\), \(z(0)= 0\), \(y(t)= \pi z(t)\), \(t\in\Delta\), is flexible, and the subspace \(M(A_0)\) is subordinated to \(B\mathbb{R}^p\) by use of the matrix \(A_0\), then the nonlinear system (1), (2) is flexible. (Recall that \(f\) is Lipschitzian!) Additional examples conclude this very interesting article.