Operator Riccati Research Articles

Linear-quadratic optimal control for abstract differential-algebraic equations

In this paper, we extend a classical approach to linear quadratic (LQ) optimal control via Popov operators to abstract linear differential-algebraic equations (ADAEs) in Hilbert spaces. To ensure existence of solutions, we assume that the underlying differential-algebraic equation has index one in the pseudo-resolvent sense. This leads to the existence of a degenerate semigroup that can be used to define a Popov operator for our system. It is shown that under a suitable coercivity assumption for the Popov operator the optimal costs can be described by a bounded Riccati operator and that the optimal control input is of feedback form. Furthermore, we characterize exponential stability of ADAEs which is required to solve the infinite horizon LQ problem.

Optimal Actuator Location for Diffusion Equation with Neumann Boundary Control

This paper is concerned with optimal actuator location for a class of diffusion equations with boundary control. The problem of optimal actuator location based on a linear quadratic cost in case of the worst initial condition is formulated. This leads to a cost involving the operator norm of the Riccati operator. The existence of optimal locations within the uniform operator norm is guaranteed. A Galerkin-based algorithm for approximating the optimal locations, and the convergence of the approximation of the optimal locations is also presented.

On the solution of the operator Riccati equations and invariant subspaces in the weighted Bergman space of the unit ball

We consider the Riccati operator equations on the weighted Bergman space A2? (Bn) of the unit ball Bn in Cn and investigate the properties of their solutions. Our discussion uses the Berezin symbols method.

Contraction analysis of Volterra integral equations via realization theory and frequency-domain methods

<p style='text-indent:20px;'>Realization theory for linear input-output operators and frequency-domain methods for the solvability of Riccati operator equations are used for the contraction analysis of a class of nonlinear Volterra integral equations in some Hilbert space. The key idea is to consider, similar to the Volterra equation, a time-invariant control system generated by an abstract ODE in some weighted Sobolev space which has the same stability properties as the Volterra equation.</p>

Interior and H∞ feedback stabilization for sabra shell model of turbulence

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models are studied for their mathematical relevance and the numerical simulations that exhibit at most resemblance with turbulent flows. One of the mathematically well‐studied shell model of turbulence is called sabra shell model. This work concerns with two important issues related to shell model, namely, feedback stabilization and robust stabilization. We first address stabilization problem related to sabra shell model of turbulence and prove that the system can be stabilized via finite dimensional controller. Thus, only finitely many modes of the shell model would suffice to stabilize the system. Later, we study robust stabilization in the presence of the unknown disturbance and corresponding control problem by solving an infinite time horizon max‐min control problem. We first prove the stabilization of the associated linearized system and characterize the optimal control in terms of a feedback operator by solving an algebraic Riccati equation. Using the same Riccati operator, we establish asymptotic stability of the nonlinear system.

Slow Decay and Turnpike for Infinite-Horizon Hyperbolic Linear Quadratic Problems

This paper is devoted to analyzing the explicit slow decay rate and turnpike in infinite-horizon linear quadratic optimal control problems for hyperbolic systems. Under suitable weak observability or controllability conditions, lower and upper bounds of the corresponding algebraic Riccati operator are proved. Then based on these two bounds, the explicit slow decay rate of the closed-loop system with Riccati-based optimal feedback control is obtained. The averaged turnpike property for this problem is also further discussed. We then apply these results to LQ optimal control problems constrained to networks of one-dimensional wave equations and also some multidimensional ones with local controls which lack a geometric control condition.

Uniqueness of the Riccati operator of the non-standard ARE of a third order dynamics with boundary control

Abstract The Moore-Gibson-Thompson [MGT] dynamics is considered. This third order in time evolution arises within the context of acoustic wave propagation with applications in high frequency ultrasound technology. The optimal boundary feedback control is constructed in order to have on-line regulation. The above requires wellposedness of the associated Algebraic Riccati Equation. The paper by Lasiecka and Triggiani (2022) recently contributed a comprehensive study of the Optimal Control Problem for the MGT-third order dynamics with boundary control, over an infinite time-horizon. A critical missing point in such a study is the issue of uniqueness (within a specific class) of the corresponding highly non-standard Algebraic Riccati Equation. The present note resolves this problem in the positive, thus completing the study of Lasiecka and Triggiani (2022) with the final goal of having on line feedback control, which is also optimal.

Sparse Grid Approximation of the Riccati Operator for Closed Loop Parabolic Control Problems with Dirichlet Boundary Control

Sparse Grid Approximation of the Riccati Operator for Closed Loop Parabolic Control Problems with Dirichlet Boundary Control

New interpretation of elliptic Boundary value problems via invariant embedding approach and Yosida regularization

The method of invariant embedding for the solutions of boundary value problems yields an equivalent formulation to the initial boundary value problems by a system of Riccati operator differential equations. A combined technique based on invariant embedding approach and Yosida regularization is proposed in this paper for solving abstract Riccati problems and Dirichlet problems for the Poisson equation over a circular domain. We exhibit, in polar coordinates, the associated Neumann to Dirichlet operator, somme concrete properties of this operator are given. It also comes that from the existence of a solution for the corresponding Riccati equation, the problem can be solved in appropriate Sobolev spaces.

On the robustness of Riccati flows to complete model misspecification

Consider the continuous-time matrix Riccati operator Ricc(Q)=AQ+QA′−QSQ+R. In this work, we consider the robustness of this operator to direct perturbations of the matrices (A, R, S) and, in particular, the flow robustness of the corresponding Riccati differential equation. For a given class of perturbation, we show that the corresponding differential equation is well defined in the sense it is bounded above and below, it has a well-defined fixed point, and it converges to this fixed point exponentially fast. Moreover, the flow of the perturbed Riccati flow is close to the nominal Riccati flow when the perturbation is small; i.e. we prove a continuity-type condition in the size of the perturbation.

Some results for operators on a model space

We investigate some problems for truncated Toeplitz operators. Namely, the solvability of the Riccati operator equation on the set of all truncated Toeplitz operators on the model space K θ = H2ΘθH2 is studied. We study in terms of Berezin symbols invertibility of model operators. We also prove some results for the Berezin number of the truncated Toeplitz operators. Moreover, we study some property for H2-functions in terms of noncyclicity of co-analytic Toeplitz operators and hypercyclicity of model operators.

Solving Symmetric Algebraic Riccati Equations with High Order Iterative Schemes

We solve symmetric algebraic Riccati equation using efficient high-order iterative schemes which improve the speed of convergence of others widely used in the literature. These high-order iterative schemes involve the Riccati operator and the first Frechet derivative. Applying these iterative schemes is equivalent to solving a fixed number of Lyapunov equations with the same matrix. This key fact makes efficient these methods. Moreover, a local convergence result of one of these high-order iterative schemes is analyzed. Finally, numerical experiments confirm the advantageous performance of these schemes.

Sensor Choice for Minimum Error Variance Estimation

A Kalman filter is optimal in that the variance of the error is minimized by the estimator. It is shown here, in an infinite-dimensional context, that the solution to an operator Riccati equation minimizes the steady-state error variance. This extends a result previously known for lumped parameter systems to distributed parameter systems. It is shown then that minimizing the trace of the Riccati operator is a reasonable criterion for choosing sensor locations. It is then shown that multiple inaccurate sensors, that is, those with large noise variance, can provide as good an estimate as a single highly accurate (but probably more expensive) sensor. Optimal sensor location is then combined with estimator design. A framework for calculation of the best sensor locations using approximations is established and sensor location as well as choice is investigated with three examples. Simulations indicate that the sensor locations do affect the quality of the estimation and that multiple low-quality sensors can lead to better estimation than a single high-quality sensor.

Fourier-splitting method for solving hyperbolic LQR problems

We consider the numerical approximation to linear quadratic regulator problems for hyperbolic partial differential equations where the dynamics is driven by a strongly continuous semigroup. The optimal control is given in feedback form in terms of Riccati operator equations. The computational cost relies on solving the associated Riccati equation and computing the optimal state. In this paper we propose a novel approach based on operator splitting idea combined with Fourier's method to efficiently compute the optimal state. The Fourier's method allows to accurately approximate the exact flow making our approach computational efficient. Numerical experiments in one and two dimensions show the performance of the proposed method.

Riccati equations and Toeplitz-Berezin type symbols on Dirichlet space of unit ball

The present paper mainly gives some applications of Berezin type symbols on the Dirichlet space of unit ball. We study the solvability of some Riccati operator equations of the form XAX + XB - CX = D related to harmonic Toeplitz operators on the Dirichlet space. Especially, the invariant subspaces of Toeplitz operators are also considered.

ОБ ОДНОМ ПРИЛОЖЕНИИ ОПЕРАТОРНОГО УРАВНЕНИЯ РИККАТИ

ОБ ОДНОМ ПРИЛОЖЕНИИ ОПЕРАТОРНОГО УРАВНЕНИЯ РИККАТИ

Min-max Game Theory for Elastic and Visco-Elastic Fluid Structure Interactions

We present the salient features of a min-max game theory developed in the context of coupled PDE's with an interface. Canonical applications include linear fluid-structure interaction problem modeled by Oseen's equations coupled with elastic waves. We shall consider two models for the structures: elastic and visco-elastic. Control and disturbance are allowed to act at the interface between the two media. The sought-after saddle solutions are expressed in a pointwise feedback form, which involves a Riccati operator; that is, an operator satisfying a suitable non-standard Riccati differential equation. Motivations, applications as well as a brief historical account are also provided.

Sub-Optimal Control for Nonlinear Heat Equations

This paper presents a successive approximation approach (SAA) designing optimal controllers for a class of nonlinear heat equations with a quadratic performance index. By using the SAA, the optimal control problem for nonlinear heat equations is transformed into a sequence of nonhomogeneous linear differential Riccati operator equations. The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoin vector sequence. By using the finite-step iteration of the nonlinear compensation sequence, we can obtain a suboptimal control law.

Notes on the Riccati operator equation in open quantum systems

A recent problem [B. Gardas, J. Math. Phys. 52, 042104 (2011)] concerning an antilinear solution of the Riccati equation is solved. We also exemplify that a simplification of the Riccati equation, even under reasonable assumptions, can lead to a not equivalent equation.

Min–max game theory and nonstandard differential Riccati equations for abstract hyperbolic-like equations

We consider the abstract dynamical framework of Lasiecka and Triggiani (2000) [1, Chapter 9], which models a large variety of mixed PDE problems (see specific classes in the Introduction) with boundary or point control, all defined on a smooth, bounded domain Ω ⊂ R n , n arbitrary. This means that the input → solution map is bounded on natural function spaces. We then study min–max game theory problem over a finite time horizon. The solution is expressed in terms of a (positive, self-adjoint) time-dependent Riccati operator, solution of a non-standard differential Riccati equation, which expresses the optimal qualities in pointwise feedback form. In concrete PDE problems, both control and deterministic disturbance may be applied on the boundary, or as a Dirac measure at a point. The observation operator has some smoothing properties.