Representation and Control of Infinite Dimensional Systems

Abstract. In this paper we consider the convergence of the infinite dimensional version of the Kleinman-Newton algorithm for solving the algebraic Riccati operator equation associated with the linear quadratic regulator problem in a Hilbert space. We establish mesh independence for this algorithm and apply the result to systems governed by delay equations. Numerical examples are presented to illustrate the results. There are two basic issues that need to be addressed in developing practical numerical approximations for control. First, it is essential that the approximation scheme leads to finite dimensional approximating Riccati equations that converge (under mesh refinement) to the solution of the infinite dimensional Riccati equation. This is a well-studied problem (see [7], [14], [26], [33], and [43]). It is now well known that to obtain norm convergence for the Riccati equation, the approximation scheme must satisfy some form of convergence, dual convergence, and uniform preservation of stabilizability and detectability under mesh refinement (see [7] and [33]). These concepts will be made more precise in section 7.1. The important point here is that many "standard" convergent approximation schemes do not satisfy all the conditions necessary for norm convergence of the Riccati operators (see [16]). If this issue is ignored when one develops an approximation scheme for control design and optimization, then the resulting numerical algorithm can fail to produce accurate and useful results (see the numerical examples in section 9). In this paper we show that these properties are also key ingredients in establishing mesh independence of Newton-type algorithms.

Section: Problem Setting and Basicsmentioning

confidence: 99%

Section: The Bounded Linear Operator X Is Called a Weak Solution Of Tmentioning

confidence: 99%

See 1 more Smart Citation

Mesh Independence of Kleinman–Newton Iterations for Riccati Equations in Hilbert Space

Burns¹,

Sachs²,

Zietsman³

2008

“…As in [7], or in [24], we can show that there exists an operator Π ∈ L(V 0 n (Ω)) satisfying Π = Π * ≥ 0 and…”

mentioning

confidence: 97%

Feedback Boundary Stabilization of the Two-Dimensional Navier--Stokes Equations

Raymond¹

2006

154

174

Abstract. We study the exponential stabilization of the linearized Navier-Stokes equations around an unstable stationary solution, by means of a feedback boundary control, in dimension 2 or 3. The feedback law is determined by solving a Linear-Quadratic control problem. We do not assume that the normal component of the control is equal to zero. In that case the state equation, satisfied by the velocity field y, is decoupled into an evolution equation satisfied by P y, where P is the so-called Helmholtz projection operator, and a quasi-stationary elliptic equation satisfied by (I − P )y. Using this decomposition we show that the feedback law can be expressed only in function of P y. In the two dimensional case we show that the linear feedback law provides a local exponential stabilization of the Navier-Stokes equations.Key words. Dirichlet control, feedback control, stabilization, Navier-Stokes equations, Oseen equations, Riccati equation AMS subject classifications. 93B52, 93C20, 93D15, 35Q30, 76D55, 76D05, 76D071. Setting of the problem. Let Ω be a bounded and connected domain in R 2 or R 3 with a regular boundary Γ, ν > 0, and consider a couple (w, χ) -a velocity field and a pressure -solution to the stationary Navier-Stokes equations in Ω:We assume that w is regular and is an unstable solution of the instationary Navier-Stokes equations. The purpose of this paper is to determine a Dirichlet boundary control u, in feedback form, localized in a part of the boundary Γ, so that the corresponding controlled system:be stable for initial values y 0 small enough in an appropriate space X(Ω). In this setting, Q ∞ = Ω×(0, ∞),in Ω, y · n = 0 on Γ , and the operator M is a restriction operator precisely defined in section 2. If we set (z, q) = (w + y, χ + p) and if u = 0, we see that (z, q) is the solution to the Navier-Stokes equationsThus y 0 is a perturbation of the stationary solution w.To study the local feedback stabilization of system (1.2), we first study the feedback stabilization of the corresponding linearized system(1.3)To stabilize this system we can look for a control u belonging either to L 2 (0, ∞; V 0 (Γ)) or to L 2 (0, ∞; V 0 n (Γ)), where V 0 (Γ) = y ∈ L 2 (Γ) | y · n, 1 H −1/2 (Γ),H 1/2 (Γ) = 0 and V 0 n (Γ) = y ∈ L 2 (Γ) | y · n = 0 on Γ .

“…For more information about null controllability and Riccati equations as well as applications to various control systems we refer to [1,2,3,4,7,8,12,13]. …”

mentioning

confidence: 99%

Null Controllability and the Algebraic Riccati Equation in Banach Spaces

Neerven¹

2004

Abstract. By a recent result of Priola and Zabczyk, a null controllable linear system y (t) = Ay(t) + Bu (t) in a Hilbert space E is null controllable with vanishing energy if and only if it is null controllable and the only positive self-adjoint solution of the associated algebraic Riccati equationis the trivial solution X = 0. In this paper we extend this result to Banach spaces with an elementary proof which uses only reproducing kernel Hilbert space techniques. We also show that null controllability with vanishing energy implies null controllability.Key words. null controllability with vanishing energy, algebraic Riccati equation, reproducing kernel Hilbert space AMS subject classifications. Primary, 93B05; Secondary, 47D06, 93B03, 93C25 DOI. 10.1137/S0363012903437058Let A be the generator of a C 0 -semigroup on a real Banach space E and let B be a bounded linear operator from a real Hilbert space H into E. The pair (A, B) is said to be null controllable with vanishing energy if for all x ∈ E and all ε > 0 there exists a time t > 0 and a function u ∈ L 2 (0, t; H) satisfying u L 2 (0,t;H) < ε such that the mild solution y u,x of the linear control problemsatisfies y u,x (t) = 0. The pair (A, B) is said to be null controllable in finite time if there exists a fixed time t 0 > 0 such that for all x ∈ E there exists a function u ∈ L 2 (0, t 0 ; H) such that the mild solution of the problem (0.1) satisfies y u,x (t 0 ) = 0. For Hilbert spaces E, Priola and Zabczyk recently proved that a pair (A, B), which is null controllable in finite time, is null controllable with vanishing energy if and only if the only positive self-adjoint solution to the algebraic Riccati equationis the trivial solution X = 0 [10]. One of the main ingredients of the proof is the fact that a certain differential Riccati equation is solved in terms of a minimal energy functional. In this paper we extend the Priola-Zabczyk result to Banach spaces with a different proof which is based on reproducing kernel Hilbert space techniques, and we *