Operator Riccati Equation Research Articles

An operator Riccati equation and reflectionless Schrodinger operators

In this paper, we study a connection between the operator Riccati equation $\displaystyle S'(x)=KS(x)+S(x)K-2S(x)KS(x), \quad x\in\mathbb{R},$ and the set of reflectionless Schr\"odinger operators with operator-valued potentials.Here $K\in \mathcal{B}(H)$, $K>0$ and $S:\mathbb{R}\to \mathcal{B}(H)$, where $\mathcal{B}(H)$ is the Banach algebra of all linear continuous operators acting in a separable Hilbert space $H$. Let $\mathscr{S}^+(K)$ be the set of all solutions $S$ of the Riccati equation satisfying the conditions $0< S(0)< I $ and $S'(0)\ge 0$, with $I$ being the identity operator in $H$. We show that every solution $S\in \mathscr{S}^+(K)$ generates a reflectionless Schr\"odinger operator with some potential $q$ that is an analytic function in the strip $\displaystyle \Pi_K:=\left\{z=x+iy \mid x,y\in\mathbb{R}, \,\, |y|<\tfrac{\pi}{2\|K\|} \right\};$ moreover, $\displaystyle \|q(x+iy)\|\le2\|K\|^2\cos^{-2}(y\|K\|), \quad (x+iy)\in\Pi_K .$

About one problem of optimal control synthesis

This paper tackles the problem of characterizing the natural class, or Riccati rule space, of solutions to a specific equation. Despite the significant theoretical and practical implications, there is limited research exploring the application of spectral decomposition of non-self-adjoint differential operators to explicitly solve this nonlinear Riccati equation. Therefore, investigating operator Riccati equations holds potential to validate the dynamic programming method and address the synthesis problem.

LQG Graphon Mean Field Games: Analysis via Graphon-Invariant Subspaces

This paper studies approximate solutions to large-scale linear quadratic stochastic games with homogeneous nodal dynamics parameters and heterogeneous network couplings within the graphon mean field game framework in [2]-[4]. A graphon time-varying dynamical system model is first formulated to study the finite and then limit problems of linear quadratic Gaussian graphon mean field games (LQG-GMFG). The Nash equilibrium of the limit problem is then characterized by two coupled graphon time-varying dynamical systems. Sufficient conditions are established for the existence of a unique solution to the limit LQG-GMFG problem. For the computation of LQG-GMFG solutions, two methods are established and employed where one is based on fixed point iterations and the other on a decoupling operator Riccati equation; furthermore, two corresponding sets of solutions are established based on spectral decompositions. Finally, a set of numerical simulations on networks associated with different types of graphons are 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>

Modeling and computation of an integral operator Riccati equation for an infinite-dimensional stochastic differential equation governing streamflow discharge

We propose a linear-quadratic (LQ) control problem of streamflow discharge by optimizing an infinite-dimensional jump-driven stochastic differential equation (SDE). Our SDE is a superposition of Ornstein-Uhlenbeck processes (supOU process), generating a sub-exponential autocorrelation function observed in actual data. The integral operator Riccati equation is heuristically derived to determine the optimal control of the infinite-dimensional system. In addition, its finite-dimensional version is derived with a discretized distribution of the reversion speed and computed by a finite difference scheme. The optimality of the Riccati equation is analyzed by a verification argument. The supOU process is parameterized based on the actual data of a perennial river. The convergence of the numerical scheme is analyzed through computational experiments. Finally, we demonstrate the application of the proposed model to realistic problems along with the Kolmogorov backward equation for the performance evaluation of controls.

Optimal convergence rates for Galerkin approximation of operator Riccati equations

AbstractIn this paper we consider the problem of determining optimal convergence rates of Galerkin approximations to infinite dimensional operator Riccati equations (OREs). Optimal rates are obtained for a class of abstract distributed parameter systems evolving in an infinite dimensional Hilbert space. These general results are then applied to systems modeled by partial differential equations that generate compact and analytic semigroups. The estimates apply to distributed control and observation of classical parabolic equations and to certain vibration problems with sufficiently strong damping. The ORE is formulated as an equivalent operator‐valued Bochner integral equation and the Brezzi–Rappaz–Raviart theorem is used to obtain convergence rates. First we establish smoothing property and bounds for the solutions of the infinite dimensional ORE. Then it is shown that, under sui\ assumptions on the coefficients and domain geometry, the hp‐finite element approximations of the classical solution converges on the order of . Furthermore, these optimal error bounds are shown to hold for the functional gains that define observer and control gain operators. We provide numerical examples that corroborate the theoretical convergence rates.

Integral Operator Riccati Equations Arising in Stochastic Volterra Control Problems

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 7 November 2019Accepted: 02 February 2021Published online: 20 April 2021Keywordsinfinite-dimensional Lyapunov equation, integral operator Riccati equation, linear-quadratic control, stochastic Volterra equationsAMS Subject Headings47G10, 49N10, 34G20Publication DataISSN (print): 0363-0129ISSN (online): 1095-7138Publisher: Society for Industrial and Applied MathematicsCODEN: sjcodc

Duality-based optimal compensator for boundary control hyperbolic PDEs system: Application to a tubular cracking reactor

This paper is devoted to the design of an optimal stabilizing compensator for a boundary control distributed parameter system that is described by a set of hyperbolic partial differential equations (PDEs). The standard reformulation of a boundary control system is adopted here to write the system under a regular infinite-dimensional linear system. A finite-dimensional boundary optimal controller is designed based on the linear quadratic technique and the corresponding operator Riccati equation. On the other hand, a Luenberger observer is designed based on the duality between the control and the estimation problems. Combination of the designed controller and observer is performed to construct a stabilizing compensator. A case study of tubular cracking chemical reactor is used to test the performances of the developed algorithm.

PDE-Based Dynamic Density Estimation for Large-Scale Agent Systems

Large-scale agent systems have foreseeable applications in the near future. Estimating their macroscopic density is critical for many density-based optimization and control tasks, such as sensor deployment and city traffic scheduling. In this paper, we study the problem of estimating their dynamically varying probability density, given the agents' individual dynamics (which can be nonlinear and time-varying) and their states observed in real-time. The density evolution is shown to satisfy a linear partial differential equation uniquely determined by the agents' dynamics. We present a density filter which takes advantage of the system dynamics to gradually improve its estimation and is scalable to the agents' population. Specifically, we use kernel density estimators (KDE) to construct a noisy measurement and show that, when the agents' population is large, the measurement noise is approximately ``Gaussian''. With this important property, infinite-dimensional Kalman filters are used to design density filters. It turns out that the covariance of measurement noise depends on the true density. This state-dependence makes it necessary to approximate the covariance in the associated operator Riccati equation, rendering the density filter suboptimal. The notion of input-to-state stability is used to prove that the performance of the suboptimal density filter remains close to the optimal one. Simulation results suggest that the proposed density filter is able to quickly recognize the underlying modes of the unknown density and automatically ignore outliers, and is robust to different choices of kernel bandwidth of KDE.

Some properties of solutions to the Riccati equations in connection with Bergman spaces

We investigate the properties of solutions to operator Riccati equations in connection with Bergman spaces. We characterize some necessary conditions for the solvability of the Riccati equation X A X + X B − C X − D = 0 on the set τ of all Toeplitz operators on the Bergman spaces A α 2 ( 𝔻 ) . This extends the results of Karaev on Hardy space.

Output Regulation of Linearized Column Froth Flotation Process

This article explores the output regulation problem of the linearized column froth flotation process consisting of collection and froth regions that are modeled by a set of linear coupled heterodirectional hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs) with time delay. Collection and froth regions are combined through PDE boundary conditions and time-delay models the interconnections between PDE boundary and ODEs representing well-stirred reactor dynamics. The linearization of the original nonlinear system is carried out to obtain a linearized PDE-ODE system. To complete the output regulation of resulting PDE-ODE systems, one challenge is the design of an observer and stabilization controller. We proposed to obtain the stabilization feedback gains and the observer injection gains by solving operator Riccati equations; in particular, these gains can also assign a specific exponential decay rate. Another challenge is the solvability of regulator equations that arise during the regulator design for the linearized PDE-ODE systems. In particular, we consider the tracking control of more general reference signals, including step, ramp, harmonic, and arbitrary polynomial signals, and novelty lies in the necessity to investigate the solvability of the corresponding regulator equations. In this article, the development of state feedback and error feedback regulators are designed and the regulator performance is demonstrated by simulation studies.

On the compactness of solutions to certain operator inequalities arising from the Likhtarnikov — Yakubovich frequency theorem

We study the compactness property of operator solutions to certain operator inequalities arising from the frequency theorem of Likhtarnikov — Yakubovich for C0-semigroups. We show that the operator solution can be described through solutions of an adjoint problem as it was previously known under some regularity condition. Thus we connect some regularity properties of the semigroup with the compactness of the operator in the general case. We also prove several results useful for checking the non-compactness of operator solutions to Lyapunov inequalities and equations, into which the operator Riccati equation degenerates in certain cases arising in applications. As an example, we apply these theorems for a scalar delay equation posed in a proper Hilbert space and show that the operator solution cannot be compact. This results are related to the author recent work on a non-local reduction principle of cocycles (non-autonomous dynamical systems) in Hilbert spaces.

Solvability of the Operator Riccati Equation in the Feshbach Case

We consider a bounded block operator matrix of the form $$ L=\left(\begin{array}{cc} A & B \\ C & D \end{array} \right), $$ where the main-diagonal entries $A$ and $D$ are self-adjoint operators on Hilbert spaces $H_{_A}$ and $H_{_D}$, respectively; the coupling $B$ maps $H_{_D}$ to $H_{_A}$ and $C$ is an operator from $H_{_A}$ to $H_{_D}$. It is assumed that the spectrum $\sigma_{_D}$ of $D$ is absolutely continuous and uniform, being presented by a single band $[\alpha,\beta]\subset\mathbb{R}$, $\alpha<\beta$, and the spectrum $\sigma_{_A}$ of $A$ is embedded into $\sigma_{_D}$, that is, $\sigma_{_A}\subset(\alpha,\beta)$. We formulate conditions under which there are bounded solutions to the operator Riccati equations associated with the complexly deformed block operator matrix $L$; in such a case the deformed operator matrix $L$ admits a block diagonalization. The same conditions also ensure the Markus-Matsaev-type factorization of the Schur complement $M_{_A}(z)=A-z-B(D-z)^{-1}C$ analytically continued onto the unphysical sheet(s) of the complex $z$ plane adjacent to the band $[\alpha,\beta]$. We prove that the operator roots of the continued Schur complement $M_{_A}$ are explicitly expressed through the respective solutions to the deformed Riccati equations.

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.

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.

Parametric delta operator Riccati equation and applications to semi-global stabilization in the presence of actuator saturation

In this paper, a parametric delta operator Riccati equation is established for low gain feedbacks of linear delta operator systems. Some properties for the parametric delta operator Riccati equation are given based on a parameter-dependent cost function. An explicit solution is also given for the delta operator parametric Riccati equation. Semi-global stabilization is described for a linear delta operator system with actuator saturation via low gain state and output feedback control laws. A numerical example is given to illustrate the effectiveness and potential for the developed techniques.

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.

Practical stabilization of delta operator systems subject to actuator saturation and disturbances

This paper focuses on practical stabilization of a null controllable delta operator system subject to actuator saturation and disturbances. Some properties of the domain of attraction for the delta operator system are shown by vector fields analysis. Practical stabilization of a planar delta operator system is given by a state feedback control law which is obtained by solving an algebraic delta operator Riccati equation. Moreover, a comprehensive controller is designed to guarantee performances of a higher order delta operator system. Finally, an aircraft example is given to illustrate the effectiveness of the proposed techniques in this paper.