Linear dynamic systems with symmetries (Q798593)

The paper considers the problem of designing an optimal linear regulator \[ \int^{\infty}_{0}[(y,Qy)+(u,Ru)]dt\to\min, \] \(\dot x=Ax+Bu,\quad y=Cx,\quad x(0)=x_ 0.\) For \(V\in Hom(H\times H)\) and R(V) denoting a vector field in Hom(H) the mapping \(V\to R(V)\) is proven to be a homomorphism of the Lie algebra \(Hom(V\times V)\) in the Lie algebra of smooth vector fields in Hom(H) (in the above, Hom(H) means a set of linear continuous mappings of H into itself). As a corollary results that each element V of the algebra \[ L(G_ S)\triangleq\{V\in Hom(H\times H):\quad V_ SV=VV_ S\} \] induces a system (in general, quadratic) of relationships \(R(V)(K_ 0)=0\) in the stabilized solution \(K_ 0\) of the Riccati matrix equation. It is also demonstrated that the algebra L(G) is ''almost isomorphic'' to the subalgebra \(DL(G_ S)\cap sp(H)\) of Hamiltonian quasidiagonal operators of the algebra \(L(G_ S)\).