Linear Quadratic Optimal Control

In this section, we shall consider a control problem for the state linear system Σ (A, B, C), where Z, U, and Y are separable Hilbert spaces, A is the infinitesimal generator of a C₀-semigroup T(t) on Z, B ∈ L(U, Z), and C ∈ L(Z, Y). In contrast with the previous chapters, we shall consider the time interval (t₀, t_e] instead of the interval [0, τ]. We recall that the state and the output trajectories of the state linear system are given by 6.1$$\begin{gathered}\begin{array}{*{20}{c}}{z\left( t \right)}& = &{T\left( {t - {t_0}} \right){z_0} + \int\limits_{{t_0}}^t {T\left( {t - s} \right)Bu\left( s \right)} ds,}\end{array} \hfill \\\begin{array}{*{20}{c}}{y\left( t \right)}& = &{Cz\left( t \right),}\end{array} \hfill \\\end{gathered}$$ where z₀ ∈ Z is the initial condition. We associate the following cost functional with the trajectories (6.1) 6.2$$J\left( {{z_0};{t_0},{t_e},u} \right) = \left\langle {z\left( {{t_e}} \right),Mz\left( {{t_e}} \right)} \right\rangle + \int\limits_{{t_0}}^{{t_e}} {\left\langle {y\left( s \right),y\left( s \right)} \right\rangle } + \left\langle {u\left( s \right),Ru\left( s \right)} \right\rangle ds,$$ where z(t) is given by (6.1) and $$u{\text{ }} \in {\text{ }}{L_2}\left( {\left[ {{t_0},{\text{ }}{t_e}} \right];U} \right).$$. Furthermore, M ∈ L(Z) is self-adjoint and nonnegative, R ∈ L(U) is coercive, that is, R is self-adjoint, and R ≥ ε I for some ε > 0 (see A.3.71).