Riccati-type Equation Research Articles

ON THE REDUCTION OF SOME GENERAL RICCATI-TYPE EQUATIONS

AbstractThe first part of the paper presents a unique reduction of some general Riccati-type differential equation to the general Bernoulli-type equation. Furthermore four special theorems on reduction of some Riccati-type differential equations are considered in the second part of the paper. A number of examples are also presented to illustrate the considered issues.

A necessary and sufficient condition for the existence of the stabilizing solution of a large class of discrete-time Riccati type equations with periodic coefficients

This paper is devoted to the study of a large class of discrete-time backward Riccati equations arising in several linear quadratic (LQ) type control problems both in the deterministic and in the stochastic frameworks. The periodic time-varying case is considered. We propose existence and uniqueness conditions for a global special solution named stabilizing solution for such equations. Beside the stabilizability condition, the criterion derived in this paper is expressed based on some suitable properties of the characteristic multipliers of a discrete-time, periodic linear equation adequately constructed using the coefficients of the given equation. Our result englobes, as particular cases, several existing results in the literature.

Inverse potentials for all ℓ channels of neutron-proton scattering using reference potential approach

Reference potential approach (RPA) has been successful in obtaining inverse potentials for weakly bound diatomic molecules using the Morse function. In this work, our goal is to construct inverse potentials for all the available ℓ-channels of np-scattering using RPA. Riccati-type phase equations for various ℓ-channels are solved using 5th order Runge-Kutta method to obtain the scattering phase shifts (SPS) in tandem with an optimisation procedure to minimize mean squared error (MSE). Interaction potentials for a total of 18 states have been constructed using only a three parameter Morse interaction model. The obtained MSE is <1% for 1 S 0, 3 P 1 and 3 D 1 channels and <2% for 1 P 1 channel and <0.1% for rest of the 14 channels. Total scattering cross-sections at various lab energies obtained using RPA are found to match well with experimental ones. A complete study of the np-scattering using RPA has been undertaken using Morse function as zeroth reference for the first time.

Dynamical large deviations of linear diffusions.

Linear diffusions are used to model a large number of stochastic processes in physics, including small mechanical and electrical systems perturbed by thermal noise, as well as Brownian particles controlled by electrical and optical forces. Here we use techniques from large deviation theory to study the statistics of time-integrated functionals of linear diffusions, considering three classes of functionals or observables relevant for nonequilibrium systems which involve linear or quadratic integrals of the state in time. For these, we derive exact results for the scaled cumulant generating function and the rate function, characterizing the fluctuations of observables in the long-time limit, and study in an exact way the set of paths or effective process that underlies these fluctuations. The results give a complete description of how fluctuations arise in linear diffusions in terms of effective forces that remain linear in the state or, alternatively, in terms of fluctuating densities and currents that solve Riccati-type equations. We illustrate these results using two common nonequilibrium models, namely, transverse diffusions in two dimensions involving a nonconservative rotating force, and two interacting particles in contact with heat baths at different temperatures.

A Comparison Estimate for Singular p-Laplace Equations and Its Consequences

Comparison estimates are an important technical device in the study of regularity problems for quasilinear possibly degenerate elliptic and parabolic equations. Such tools have been employed indispensably in many papers of Mingione, Duzaar-Mingione, and Kuusi-Mingione, etc. on certain measure datum problems to obtain pointwise bounds for solutions and their full or fractional derivatives in terms of appropriate linear or nonlinear potentials. However, a comparison estimate for $p$-Laplace type elliptic equations with measure data is still unavailable in the strongly singular case $1< p\leq \frac{3n-2}{2n-1}$, where $n\geq 2$ is the dimension of the ambient space. This issue will be completely resolved in this work by proving a comparison estimate in a slightly larger range $1<p<3/2$. Applications include a `sublinear' Poincar\'e type inequality, pointwise bounds for solutions and their derivatives by Wolff's and Riesz's potentials, respectively. Some global pointwise and weighted estimates are also obtained for bounded domains, which enable us to treat a quasilinear Riccati type equation with possibly sublinear growth in the gradient.

An addendum to the problem of zero-sum LQ stochastic mean-field dynamic games

In this paper, we first address a linear quadratic mean-field game problem with a leader–follower structure. By adopting a Riccati-type approach, we show how one can obtain a state-feedback representation of the pairs of strategies which achieve an open-loop Stackelberg equilibrium in terms of the global solutions of a system of coupled matrix differential Riccati-type equations. In the second part of this paper, we obtain necessary and sufficient conditions for the solvability of the involved coupled generalized Riccati equations.

Deterministic optimal control for discrete-time systems with multiplicative noises and random coefficients

This paper studies a linear-quadratic optimal control problem for discrete-time systems with multiplicative noise and random coefficients. Motivated by fiscal policy planning problems, where deterministic policies are required, this paper aims to find a deterministic optimal controller. The challenge is to seek a solution to forward and backward stochastic difference equations with structural inconsistency caused by random and deterministic terms. We derive necessary and sufficient conditions to solve this optimal control problem by employing a decoupling method. We also present an explicit expression of the optimal controller based on coupled stochastic Riccati-type difference equations.

Indefinite Backward Stochastic Linear-Quadratic Optimal Control Problems

This paper is concerned with a backward stochastic linear-quadratic (LQ, for short) optimal control problem with deterministic coefficients. The weighting matrices are allowed to be indefinite, and cross-product terms in the control and state processes are presented in the cost functional. Based on a Hilbert space method, necessary and sufficient conditions are derived for the solvability of the problem, and a general approach for constructing optimal controls is developed. The crucial step in this construction is to establish the solvability of a Riccati-type equation, which is accomplished under a fairly weak condition by investigating the connection with forward stochastic LQ optimal control problems.

Existence and regularity estimates for quasilinear equations with measure data : the case 1 &lt; p ≤ (3n − 2)∕(2n − 1)

We obtain existence and global regularity estimates for gradients of solutions to quasilinear elliptic equations with measure data whose prototypes are of the form $-{\rm div} (|\nabla u|^{p-2} \nabla u)= \delta\, |\nabla u|^q +\mu$ in a bounded main $\Om\subset\RR^n$ potentially with non-smooth boundary. Here either $\delta=0$ or $\delta=1$, $\mu$ is a finite signed Radon measure in $\Omega$, and $q$ is of linear or super-linear growth, i.e., $q\geq 1$. Our main concern is to extend earlier results to the strongly singular case $1<p\leq \frac{3n-2}{2n-1}$. In particular, in the case $\delta=1$ which corresponds to a Riccati type equation, we settle the question of solvability that has been raised for some time in the literature.

Solution to Forward–Backward Stochastic Differential Equations With Random Coefficients and Application to Deterministic Optimal Control

We address forward–backward stochastic differential equations (FBSDEs) with random coefficients. Differently from previous approaches, we consider FBSDEs with coefficients that are random and correlated with forward and backward variables. In addition, expectation terms appear in the forward equation, and this makes challenging to solve coupled FBSDEs, e.g., those coming from corporations’ fiscal policy planning. Using a discretization approach, an analytic solution to the FBSDEs is given in terms of coupled stochastic Riccati-type differential equations. We also apply our results to optimal deterministic control problem for Itô stochastic systems with both random and deterministic coefficients. Our results show that the discretization approach is a powerful tool to decouple complex FBSDEs.

The Covariance Extension Equation: A Riccati-Type Approach to Analytic Interpolation

Analytic interpolation problems with rationality and derivative constraints are ubiquitous in systems and control. This article provides a new method for such problems, both in the scalar and matrix case, based on a nonstandard Riccati-type equation. The rank of the solution matrix is the same as the degree of the interpolant, thus providing a natural approach to model reduction. A homotopy continuation method is presented and applied to some problems in modeling and robust control. We also address a question on the positive degree of a covariance sequence originally posed by Kalman.

Mixed optimal control for discrete-time stochastic systems with random coefficients

This paper is concerned with the problem of mixed optimal control for discrete-time systems with random coefficients where “mixed” means that there are two controllers: the deterministic controller and the random controller. Different from previous works, random coefficients and controllers with different information structures arise simultaneously in the optimal control problem, which lead to large difficulty. Using decoupling technique to solve the forward and backward stochastic difference equations, the necessary and sufficient conditions for the solvability of optimal control problem are given. An explicit expression of the optimal control strategies is derived based on a stochastic Riccati-type equation, which is new to the best of knowledge. Finally, a pension management problem is solved with the obtained theoretical results.

Optimal control and stabilization for NCSs with multiple memoryless channels and hold‐input compensation

This paper investigates the optimal control problem for networked control systems over multiple Markovian packet loss channels under hold-input compensation strategy. In this situation, the actuator receives the control signals from the controller via the individual channels. The system is transformed into a Markovian jump linear system by introducing a new Markovian jump variable. By giving a new maximum principle for such a system, a coupled difference Riccati-type equation is developed, which in turn gives necessary and sufficient conditions for the solvability of optimal control problems in the finite horizon. Furthermore, for the infinite horizon case, the Markovian jump linear system is mean square stabilizable if and only if a set of coupled algebraic Riccati-type equation has a particular solution. Finally, numerical simulations are presented to illustrate the control effects of the proposed optimal controllers.

On the Riccati dynamics of 2D EulerPoisson equations with attractive forcing

The Euler–Poisson (EP) system describes the dynamic behaviour of many important physical flows. In this work, a Riccati system that governs pressureless two-dimensional EP equations is studied. The evolution of divergence is governed by the Riccati type equation with several nonlinear/nonlocal terms. Among these, the vorticity accelerates divergence while others further amplify the blow-up behaviour of a flow. The growth of these blow-up amplifying terms are related to the Riesz transform of density, which lacks a uniform bound makes it difficult to study global solutions of the multi-dimensional EP system. We show that the Riccati system can afford to have global solutions, as long as the growth rate of blow-up amplifying terms is not higher than exponential, and admits global smooth solutions for a large set of initial configurations. To show this, we construct an auxiliary system in 3D space and find an invariant space of the system, then comparison with the original 2D system is performed. Some numerical examples are also presented.

Bäcklund transformation, Lax pair, infinite conservation laws and exact solutions to a generalized (2 + 1)-dimensional equation

In this paper, the integrability of a generalized ([Formula: see text])-dimensional equation is investigated by using the Bell polynomials and Hirota bilinear method. By appropriate transformation, the bilinear equation is constructed. Some soliton solutions are yielded via the Hirota bilinear method. Then the evolution and collision of the soliton solutions are discussed. Starting from the two-field condition, the bilinear Bäcklund transformation is derived. Linearization of the [Formula: see text]-polynomials-typed Bäcklund transformation leads to the Lax pair. Besides, infinite conservation laws are derived by constructing the Riccati-type equation and devergence-type equation.

Decentralized control for multi-sensors networked systems with different transmission delays and packet dropouts

In this paper, the linear quadratic (LQ) optimal decentralized control and stabilization problems are investigated for multi-sensors networked control systems (MSNCSs) with multiple controllers of different information structure. Specifically, for a MSNCS, in view of the packet dropouts and the transmission delays, each controller may access different information sets. To begin with, the sufficient and necessary solvability conditions for the LQ decentralized control problems are developed. Consequently, for the purpose of deriving the optimal decentralized control strategy, an innovative orthogonal decomposition method is proposed to decouple the forward and backward stochastic difference equations (FBSDEs) from the maximum principle. In the following, we show that the optimal decentralized controller can be calculated according to a set of Riccati-type equations. Finally, a stabilizing controller is derived for the stabilization problem.

A suboptimal control of linear time-delay problems via dynamic programming

Abstract We study a class of infinite horizon optimal control problems with a state delay, and investigate the dynamic programming approach which leverages the sufficient optimality conditions and provides a closed-loop solution. Importantly, the well-known Lyapunov–Krasovskii functional is applied to relate the solution of the problem to the solution of a set of three Riccati-type matrix differential equations. We then present an analytic-based approach to solve the resultant equations and subsequently provide a suboptimal closed-loop solution for the considered problem. We prove the uniform convergence of the proposed approach and show that the presented closed-loop system is asymptotically stable in the Lyapunov sense. Furthermore, the observability of the linear time-delay system is discussed and proved. Finally, numerical examples illustrate the efficiency of the proposed method.

Vacuum and singularity formation problem for compressible Euler equations with general pressure law and time-dependent damping

In this paper, the vacuum and singularity formation problem are considered for the compressible Euler equations with general pressure law and time-dependent damping. Firstly, the lower bound estimates of density for arbitrary classical solutions is shown for some kind of pressure functions. Then, three sufficient conditions, under which the classical solutions must break down in finite time, are shown by delicate analysis of decoupled Riccati type equations. The assumptions on pressure function automatically satisfied for gas dynamics with γ-law. Furthermore, our results have no limits on the size of the solutions or the positive/monotonicity on the initial Riemann invariants.

Exponential stabilization on infinite dimensional system with impulse controls

This paper studies the exponential stabilization on the infinite dimensional system with impulse controls, where the impulse instants appear periodically. The first main result shows that the exponential stabilizability of the control system with a periodic feedback law is equivalent to one kind of weak observability inequalities. The second main result presents that, in the setting of a discrete LQ problem, the exponential stabilizability of control system with a periodic feedback law is equivalent to the solvability of an algebraic Riccati-type equation which was built up in Qin et al. (2021) [18] for finite dimensional system. As an application, some sufficient and necessary condition for the exponential stabilization of an impulse controlled system governed by coupled heat equations is given.

Distributed filtering based on Cauchy-kernel-based maximum correntropy subject to randomly occurring cyber-attacks

This paper is concerned with the distributed filtering issue under the Cauchy-kernel-based maximum correntropy for large-scale systems subject to randomly occurring cyber-attacks in non-Gaussian environments. The considered cyber-attacks are hybrid and consist of both denial-of-service attacks and deception attacks. The weighted Cauchy kernel-based maximum correntropy criterion instead of the traditional minimum variance is put forward to evaluate the filtering performance against non-Gaussian noises as well as cyber-attacks. Based on the matrix decomposition and the fixed-point iterative update rules, the desired filter gain related with a set of Riccati-type equations is obtained to achieve the optimal filtering performance. Then, an improved version only dependent on the local information and neighboring one-step prediction is developed to realize the distributed implementation. Furthermore, the convergence of the developed fixed-point iterative algorithm is addressed via the famous Banach fixed-point theorem. Finally, a standard IEEE 39-bus power system is utilized to show the merit of the proposed distributed filtering algorithm in the presence of cyber-attacks and non-Gaussian noises.