Optimal control problems with hybrid quadratic criteria (Q1097507)

Given the linear distributed parameter system \[ (*)\quad x(t)=T(t)x_ 0+\int^{t}_{0}T(t-\sigma)Bd\sigma,\quad t\in [0,t_ f], \] X and U are the real Hilbert spaces of state and control; T(t), \(t\geq 0\) is a \(C_ 0\)-semigroup of linear operators in \({\mathcal L}(X)\); \(B\in {\mathcal L}(U;X)\); \(x_ 0\in X\) is fixed; \(u\in {\mathcal U}\); \(x(.)\in C([0,t_ f];x)\); \(t_ f>0\) is finite. The optimal control should satisfy the following criterion: \[ \inf_{u\in {\mathcal U}}\{\sum^{N}_{i=1}<Q_ ix(t_ i),x(t_ i)>+\int^{t_ f}_{0}[\Omega (.)+<Ru(t),u(t)>]dt, \] where \(\Omega (.)=<Qx(t),x(t)>\), \(Q_ i\) \((i=1,2,...,N)\) and Q are non-negative self-adjoint operators in \({\mathcal L}(x)\); \(R\in {\mathcal L}(U)\). Each integral has to be understood as a strictly Bochner one. Expressions of feedback operators for optimal control are given in the following three cases: a) \(\Omega\) (.)\(\equiv 0\), the measurements of state x(t) is performed in real time; b) \(\Omega\) (.)\(\equiv 0\), the state \(x(t_ i)\) measurement is performed in discrete time \(t_ i\leq t<t_{i+1}\); c) the state \(x(t_ i)\) is performed in discrete time \(t_ i\leq t<t_{i+1}\), \(\Omega (.)=<Qx(t),x(t)>.\) The problem discussed in the paper has quite a practical meaning, because the minimization of control energy and state measurement in discrete time are frequently the technical requirements when synthesizing the optimal control.