Abstract
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, te] instead of the interval [0, τ]. We recall that the state and the output trajectories of the state linear system are given by
where z0 ∈ Z is the initial condition. We associate the following cost functional with the trajectories (6.1)
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).
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© 1995 Springer-Verlag New York, Inc.
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Curtain, R.F., Zwart, H. (1995). Linear Quadratic Optimal Control. In: An Introduction to Infinite-Dimensional Linear Systems Theory. Texts in Applied Mathematics, vol 21. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4224-6_6
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DOI: https://doi.org/10.1007/978-1-4612-4224-6_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-8702-5
Online ISBN: 978-1-4612-4224-6
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