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6 Chapter 1. Introduction
and in related factorization problems (see [ALT01]) and the references therein). Finally, let us point out that, during the past three decades, a significant number of papers addressed the computation of the spectral factorization. More details on numerical aspects and on different approaches to this problem can be found in [OaVa00]. The Lyapunov–Riccati transformation has for example been used in [SGS92] in order to exploit the reduced-order slow and fast subsystem of a singular perturbed linear control system.
In [SuGa91] the Lyapunov–Riccati transformation is applied as an essential tool in the study of continuous and discrete weakly coupled systems (see also [GPS90] for further details). A slightly different approach has been used in [AgGa93] for the investigation of weakly coupled bilinear control systems with a quadratic cost functional.
Next we briefly indicate an application to linear boundary value problems.
1.2 Boundary value problems
In variational calculus or in control theory and game theory, necessary con- ditions for an equilibrium are determined by the solvability of a linear boundary value problem in the time interval [t0,tf]. Also in wave propagation, invariant imbedding, spectral and scattering theory such boundary value problems ap- pear which can be solved by using appropriate Riccati equations (see [Denm86], [CaMo00], [ClGe02]). Moreover, similar boundary value problems appear in the theory of transport processes (see [BUV73]) and the oscillation theory of second order differential systems [KaKw86], [Kwon83]. They take the form
d x M11 M12x m n dt ψ = M21 M22 ψ ,x(t)∈R ,ψ(t)∈R ,
x(t0) = x0, ψ(tf) = Kfx(tf), Kf ∈ Rn×m. In control theory the matrix
M12 (m+n)×(m+n) M (t) ∈ R
22
(1.10)
M(t) = M 21
M11
turns out to be Hamiltonian which is indeed a rather specific situation. A solu- tion of this boundary value problem may once again be related to the solvability of an associated Riccati differential equation.
Theorem 1.2.1. If the Riccati differential equation
W˙ = M12 + M11W − WM22 − WM21W, W(t0) = 0, W(t) ∈ Rm×n (1.11)
admits a solution in [t0 , tf ] and if
det(Kf W (tf ) − In ) ̸= 0,