Riccati-type Equation Research Articles

Decoupled Stein iterations to the discrete-time generalized Riccati equations

The authors investigate the numerical solution of a set of discrete-time generalised Riccati equations. The class of discrete-time non-linear equations involves in various control problems for discrete-time stochastic systems and it can be considered as an important tool for solving optimisation control for such type systems. A new procedure for computing the maximal solution and the stabilising solution is proposed by Dragan et al. (‘Iterative algorithm to compute the maximal and stabilising solutions of a general class of discrete-time Riccati-type equations’, Int. J. Control, 2010, 83, (4), pp. 837–847). In this study, the authors introduce a new iterative procedure based on the solution of a Stein matrix equation for computing the maximal and the stabilising solution. The convergence properties of the new iteration are proved. Sufficient conditions for computing the maximal solution and the stabilising solution are derived. Finally, some numerical examples are presented to illustrate the feasibility of the proposed algorithm.

The weighted Stieltjes inequality and applications

AbstractLet 1 < p ≤ q < ∞. Inspired by some results concerning characterization of weighted Hardy type inequalities, where the equivalence of four scales of integral conditions was proved, we use related ideas to find some new equivalent scales of integral conditions related to the Stieltjes transform. By applying our result to weighted inequalities for the Stieltjes transform we obtain four new scales of conditions for characterization of this inequality. We also derive a new characterization for the solvability of a Riccati type equation and show via our new results that this characterization can be done in infinite many ways via our four scales of equivalent conditions.

Delayed Stochastic Linear‐Quadratic Control Problem and Related Applications

We discuss a quadratic criterion optimal control problem for stochastic linear system with delay in both state and control variables. This problem will lead to a kind of generalized forward‐backward stochastic differential equations (FBSDEs) with Itô’s stochastic delay equations as forward equations and anticipated backward stochastic differential equations as backward equations. Especially, we present the optimal feedback regulator for the time delay system via a new type of Riccati equations and also apply to a population optimal control problem.

A model of competition

A competition model is described by a nonlinear first-order <i>differential</i> equation (of Riccati type). Its solution is then used to construct a <i>functional</i> equation in two variables (admitting essentially the same solution) and several itera

Stochastic Singular Optimal Control Problem with State Delays: Regularization, Singular Perturbation, and Minimizing Sequence

An optimal control problem with quadratic cost functional for a linear stochastic system with pointwise and distributed time delays in the state variable is considered. The case where the cost functional does not contain a control cost is treated. The latter means that the problem under consideration is a singular optimal control problem. This control problem is associated with a new optimal control problem for the same equation of dynamics. The cost functional in this new problem is the sum of the original cost functional and an integral of the square of the control with a small positive weighting coefficient. Thus, the new problem is a regular optimal control problem. Moreover, it is a cheap control problem. By using the control optimality conditions, the solution of this problem is reduced to solution of the set of four equations: one Riccati-type matrix ordinary differential equation, two Riccati-type matrix partial differential equations of the first order, and one trivial scalar ordinary differential equation. This set of equations is with deviating arguments, and it is singularly perturbed. An asymptotic solution of this set of equations is constructed and justified. Based on this asymptotic solution, an asymptotically suboptimal state-feedback control is constructed for the cheap control problem. Finally, it is shown that this control constitutes a minimizing sequence for the original problem. The infimum of the cost functional in the original optimal control problem also is obtained. An illustrative example is presented.

Feedback Nash Strategy for games with player-dependent time horizons

In this paper, differential games in discrete time are considered. In such a setting it is generally assumed that the time horizon characterizing the criteria of all players is the same. We consider here that the time horizon of the criteria depends on the player, that is typically there is long and short time horizon. For the Nash strategy with a feedback information structure applied to a linear quadratic game, the sufficient conditions of existence are introduced and formulated into a new kind of coupled Riccati-type equations mixing an algebraic and a time-dependent structures. An iterative algorithm is provided to solve them. Illustrations allow to show the efficiency of the proposed algorithm and consolidate the discussion.

The Linear Quadratic Regulator Problem for a Class of Controlled Systems Modeled by Singularly Perturbed Itô Differential Equations

This paper discusses an infinite-horizon linear quadratic (LQ) optimal control problem involving state- and control-dependent noise in singularly perturbed stochastic systems. First, an asymptotic structure along with a stabilizing solution for the stochastic algebraic Riccati equation (ARE) are newly established. It is shown that the dominant part of this solution can be obtained by solving a parameter-independent system of coupled Riccati-type equations. Moreover, sufficient conditions for the existence of the stabilizing solution to the problem are given. A new sequential numerical algorithm for solving the reduced-order AREs is also described. Based on the asymptotic behavior of the ARE, a class of $O(\sqrt{\varepsilon})$ approximate controller that stabilizes the system is obtained. Unlike the existing results in singularly perturbed deterministic systems, it is noteworthy that the resulting controller achieves an $O(\varepsilon)$ approximation to the optimal cost of the original LQ optimal control problem. As a result, the proposed control methodology can be applied to practical applications even if the value of the small parameter $\varepsilon$ is not precisely known.

Difference and differential equations for deformed Laguerre–Hahn orthogonal polynomials on the unit circle

Sequences of orthogonal polynomials on the unit circle whose Carathéodory function satisfies a Riccati-type differential equation with polynomial coefficients are studied. We deduce discrete Lax equations which lead to difference equations for the corresponding sequences of reflection parameters, and we analyze the continuous differential equations that arise when deformations through a dependence on a parameter t occur.

Smooth solutions and singularity formation for the inhomogeneous nonlinear wave equation

We study the nonlinear inhomogeneous wave equation in one space dimension: v t t − T ( v , x ) x x = 0 . By constructing some “decoupled” Riccati type equations for smooth solutions, we provide a singularity formation result without restrictions on the total variation of the data, which generalizes earlier singularity results of Lax and the first author. We apply these results to compressible Euler flows with a general pressure law and elasticity in an inhomogeneous medium.

Wave front-ray synthesis for solving the multidimensional quantum Hamilton-Jacobi equation

A Cauchy initial-value approach to the complex-valued quantum Hamilton-Jacobi equation (QHJE) is investigated for multidimensional systems. In this approach, ray segments foliate configuration space which is laminated by surfaces of constant action. The QHJE incorporates all quantum effects through a term involving the divergence of the quantum momentum function (QMF). The divergence term may be expressed as a sum of two terms, one involving displacement along the ray and the other incorporating the local curvature of the action surface. It is shown that curvature of the wave front may be computed from coefficients of the first and second fundamental forms from differential geometry that are associated with the surface. Using the expression for the divergence, the QHJE becomes a Riccati-type ordinary differential equation (ODE) for the complex-valued QMF, which is parametrized by the arc length along the ray. In order to integrate over possible singularities in the QMF, a stable and accurate Möbius propagator is introduced. This method is then used to evolve rays and wave fronts for four systems in two and three dimensions. From the QMF along each ray, the wave function can be easily computed. Computational difficulties that may arise are described and some ways to circumvent them are presented.

Cauchy matrix for linear almost periodic systems and some consequences

A novel method to construct the fundamental matrix for a linear almost periodic system is proposed, provided that the diagonal terms satisfy an average separation condition and the off-diagonal coefficients are L ∞ -small. The idea is to transform the system in a set of Riccati type equations and use exponential dichotomy and its consequences. It is shown that the method yields easy computation procedures with simple and direct conditions depending on the coefficients. Finally, our result enables us to obtain: (i) explicit almost periodic matrices Q ( t ) , Q − 1 ( t ) and Q ′ ( t ) , which diagonalize the original system and (ii) sufficient conditions for the stability. Two illustrative examples are shown.

Suboptimal control of feedback-linearizable nonlinear plant

An optimal control problem is formulated for a class of nonlinear systems for which there exists a coordinate representation (diffeomorphism) transforming the original system into a system with a linear main part and a nonlinear feedback. In this case the coordinate transformation significantly changes the form of original quadratic functional. The penalty matrices become dependent on the system state. The linearity of the structure of the transformed system and the quadratic functional make it possible to pass over from the Hamilton-Jacoby-Bellman equation to the Riccati-type equation with state-dependent parameters upon the control synthesis. Note that it is impossible to solve the obtained form of Riccati equation analytically in the general case. It is necessary to approximate the solution; this approximation is realized by numerical methods using symbolic computer packages or interpolation methods. In the latter case, it is possible to obtain the suboptimal control. The presented example illustrates the application of the proposed control method for the feedback-linearizable nonlinear system.

An anticipative linear filtering equation

In the classical Kalman–Bucy filter and in the subsequent literature so far, it has been assumed that the initial value of the signal process is independent of both the noise of the signal and of the noise of the observations. The purpose of this paper is to prove a filtering equation for a linear system where the (normally distributed) initial value X 0 of the signal process X t has a given correlation function with the noise (Brownian motion B t ) of the observation process Z t . This situation is of interest in applications to insider trading in finance. We prove a Riccati type equation for the mean square error S ( t ) : = E [ ( X t − X ˆ t ) 2 ] ; 0 ≤ t ≤ T , where X ˆ t is the filtered estimate for X t . Moreover, we establish a stochastic differential equation for X ˆ t based on S ( t ) . Our method is based on an enlargement of filtration technique, which allows us to put the anticipative linear filter problem into the context of a non-anticipative two-dimensional linear filter problem with a correlation between the signal noise and the observation noise.

Quasilinear Riccati Type Equations and Quasiminimizers

Abstract Solutions u of the equation -∇ ・ A(x, ∇u) = b(x)|∇u|q are studied in the subcritical case p - 1 ≤ q ≤ p/n + p - 1, q &lt; p &lt; n. Here A(x, h) satisfies A(x, h) ・ h ≈ |h|p and b a bounded function in an open set Ω ⊂ Rn. It is shown that the solutions u are local quasiminimizers in Ω. From the quasiminimizing property several conclusions on the behavior of the solutions u can be derived. In particular, the method provides an easy solution to the uniqueness problem in the class W0 1,p(Ω).

Differential equations where the derivative is taken with respect to a measure

This paper looks at ordinary differential equations (DE) containing the derivative of the unknown functions with respect to a measure which is continuous with respect to the Lebesgue measure. It is shown that the Cauchy problem for a linear normal system of DE with a -derivative is uniquely solvable. A necessary and sufficient condition is obtained for the solvability of an equation of Riccati type with a -derivative. It is related to a boundary-value problem for a linear system of DE. Using this condition a necessary and sufficient condition is obtained for a Volterra factorization to exist for linear operators that differ from the identity by an integral operator that is completely continuous in the space , .Bibliography: 12 titles.

On Efficient Method for System of Fractional Differential Equations

The present study introduces a new version of homotopy perturbation method for the solution of system of fractional-order differential equations. In this approach, the solution is considered as a Taylor series expansion that converges rapidly to the nonlinear problem. The systems include fractional-order stiff system, the fractional-order Genesio system, and the fractional-order matrix Riccati-type differential equation. The new approximate analytical procedure depends only on two components. Comparing the methodology with some known techniques shows that the present method is relatively easy, less computational, and highly accurate.

Quasilinear Riccati Type Equations with Super-Critical Exponents

We establish explicit criteria of solvability for the quasilinear Riccati type equation − Δ p u = |∇u| q + ω in a bounded 𝒞1 domain Ω ⊂ ℝ n , n ≥ 2. Here Δ p , p > 1, is the p-Laplacian, q is in the supper critical range q > p, and the datum ω is a measure. Our existence criteria are given in the form of potential theoretic or geometric estimates that are sharp when ω is nonnegative and compactly supported in Ω. Our existence results are new even in the case dω =f dx where f belongs to the weak Lebesgue space . Moreover, our methods allow the treatment of more general equations where the principal operators may have discontinuous coefficients. As a consequence of the solvability results, a characterization of removable singularities for the corresponding homogeneous equation is also obtained.

Iterative algorithm to compute the maximal and stabilising solutions of a general class of discrete-time Riccati-type equations

In this article an iterative method to compute the maximal solution and the stabilising solution, respectively, of a wide class of discrete-time nonlinear equations on the linear space of symmetric matrices is proposed. The class of discrete-time nonlinear equations under consideration contains, as special cases, different types of discrete-time Riccati equations involved in various control problems for discrete-time stochastic systems. This article may be viewed as an addendum of the work of Dragan and Morozan (Dragan, V. and Morozan, T. (2009), ‘A Class of Discrete Time Generalized Riccati Equations’, Journal of Difference Equations and Applications, first published on 11 December 2009 (iFirst), doi: 10.1080/10236190802389381) where necessary and sufficient conditions for the existence of the maximal solution and stabilising solution of this kind of discrete-time nonlinear equations are given. The aim of this article is to provide a procedure for numerical computation of the maximal solution and the stabilising solution, respectively, simpler than the method based on the Newton–Kantorovich algorithm.

Limit Behavior of a Linear-Quadratic Optimal Control Problem for Singularly Perturbed Systems with Small Delays

A finite horizon quadratic cost optimal control problem for a linear singularly perturbed differential system with small point-wise and distributed time delays in the state variable is considered. By a linear transformation, this problem is converted to an equivalent linear-quadratic optimal control problem for a much simpler system. For this new problem, an asymptotic solution of the Riccati-type matrix differential equation, arising in the control optimality condition, is constructed and justified. Based on this solution, a limit behavior of the optimal value of the cost functional in the transformed problem is obtained. Using this result, a limit behavior of the optimal value of the cost functional in the original problem is established.

An Anticipative Linear Filtering Equation

In the classical Kalman-Bucy filter and in the subsequent literature so far, it has been assumed that the initial value of the signal process is independent of both the noise of the signal and of the noise of the observations.The purpose of this paper is to prove a filtering equation for a linear system where the (normally distributed) initial value X0 of the signal process Xt has a given correlation function with the noise (Brownian motion Bt) of the observation process Zt. This situation is of interest in applications to insider trading in finance. We prove a Riccati type equation for the mean square error S(t):= E[(Xt - ^Xt)**2]; 0