A control operator and some of its applications (Q802132)

This paper develops a conjugate gradient method algorithm for solving the optimal control problem of the linear system \(x'(t)=Cx(t)+Du(t)\), \(0\leq t\leq \sigma\) with an integral quadratic cost \[ \int^{\sigma}_{0}(x^ T(t)Qu(t)+u^ T(t)Ru(t)\,dt, \] where \(x\in {\mathbb R}^ n\), \(u\in {\mathbb R}^ q\), \(C\in L({\mathbb R}^ n)\), \(D\in L({\mathbb R}^ q,{\mathbb R}^ n)\) and \(Q\) and \(R\) are symmetric, positive definite, square matrices of dimensions \(n\) and \(q\), respectively. The control operator, proposed in this paper, is defined by \(A\in L(K)\), \(K=H^ 1([0,\sigma];{\mathbb R}^ n)\times L^ 2([0,\sigma];{\mathbb R}^ q)\) and \[ \langle z,Az\rangle_ K=J(x,u,\mu)=\int^{\sigma}_{0}(x^ T(t)Qx(t)+u^ T(t)Ru(t))\,dt+\mu \int^{\sigma}_{0}\| x'(t)-Cx(t)- Du(t)\|^ 2\,dt, \] where \(\mu >0\) is the penalty constant. If \(A\) can be determined explicitly, then the conjugate gradient method algorithm is applicable. The construction of the control operator \(A\) is stated by several equalities. Thus, an extended conjugate gradient method algorithm can be working. Some further applications of the control operator are discussed, too.