Riccati-type Differential Equations 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.

Robust Solution of the Multi-Model Singular Linear-Quadratic Optimal Control Problem: Regularization Approach

We consider a finite horizon multi-model linear-quadratic optimal control problem. For this problem, we treat the case where the problem’s functional does not contain a control function. The latter means that the problem under consideration is a singular optimal control problem. To solve this problem, we associate it with a new optimal control problem for the same multi-model system. The functional in this new problem is the sum of the original functional and an integral of the square of the Euclidean norm of the vector-valued control with a small positive weighting coefficient. Thus, the new problem is regular. Moreover, it is a multi-model cheap control problem. Using the solvability conditions (Robust Maximum Principle), the solution of this cheap control problem is reduced to the solution of the following three problems: (i) a terminal-value problem for an extended matrix Riccati type differential equation; (ii) an initial-value problem for an extended vector linear differential equation; (iii) a nonlinear optimization (mathematical programming) problem. We analyze an asymptotic behavior of these problems. Using this asymptotic analysis, we design the minimizing sequence of state-feedback controls for the original multi-model singular optimal control problem, and obtain the infimum of the functional of this problem. We illustrate the theoretical results with an academic example.

Integrable differential systems for deformed Laguerre–Hahn orthogonal polynomials

Our work studies sequences of orthogonal polynomials of the Laguerre–Hahn class, whose Stieltjes functions satisfy a Riccati type differential equation with polynomial coefficients, which are subject to a deformation parameter t. We derive systems of differential equations and give Lax pairs, yielding nonlinear differential equations in t for the recurrence relation coefficients and Lax matrices of the orthogonal polynomials. A specialisation to a non semi-classical case obtained via a Möbius transformation of a Stieltjes function related to a deformed Jacobi weight is studied in detail, showing this system is governed by a differential equation of the Painlevé type P. The particular case of P arising here has the same four parameters as the solution found by Magnus (1995 J. Comput. Appl. Math. 57 215–37) but differs in the boundary conditions.

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.

"A spectral criterion for the existence of the stabilizing solution of a class of Riccati type differential equations with periodic coefficients"

"In this paper, we investigate the existence and uniqueness of a stabilizing solution to a periodic backward nonlinear differential equation. This class of nonlinear equations includes as special cases many of the continuous-time Riccati equations arising both in deterministic and stochastic linear quadratic (LQ) type control problems."

Observing the epidemiological SIR model on COVID-19 pandemic data

This article shows that in the period January 22-June 04, 2020, the combined data set of cumulative recoveries and deaths from the current coronavirus COVID-19 pandemic falls on the Kermack and McKendrick approximated solution of the epidemiological {\sir} contagious disease model. Then, as an original contribution of this work, based on the knowledge of the infectious period of any epidemic, a methodology is presented that helps to find numerical solutions of the full {\sir} model that falls on the observed data of the epidemic in case it could be described by the {\sir} model. The methodology is first illustrated by finding a solution of the {\sir} model that falls on the epidemic data of the Bombay plague of 1905-06 analyzed by Kermack and McKendrick. After that, the methodology is applied on analyzing the previously considered coronavirus COVID-19 pandemic data set. Moreover, since the Kermack and McKendrick approximated solution of the {\sir} model comes from solving a Riccati type differential equation, commonly found when studying (in introductory physics courses) the vertical motion of objects on a resistive medium, enough details are given in the article so the epidemiological {\sir} model can be used as an additional example for enhancing and enriching the undergraduate curriculum Physics courses for Biology, Life Sciences, Medicine and/or Computational Modeling.

Numerical Approximation of Riccati Type Differential Equations

Numerical Approximation of Riccati Type Differential Equations

H2/H∞ Control for Stochastic Jump-Diffusion Systems with Markovian Switching

In this paper, a stochastic H2/H∞ control problem is investigated for Poisson jump-diffusion systems with Markovian switching, which are driven by a Brownian motion and a Poisson random measure with the system parameters modulated by a continuous-time finite-state Markov chain. A stochastic jump bounded real lemma is proved, which reveals that the norm of the perturbation operator below a given threshold is equivalent to the existence of a global solution to a parameterized system of Riccati type differential equations. This result enables the authors to obtain sufficient and necessary conditions for the existence of H2/H∞ control in terms of two sets of interconnected systems of Riccati type differential equations.

Closed form parametric solutions of nonlinear Abel-type and Riccati-type spacecraft relative motion

In comparison to the conventional approach of using Cartesian coordinates to describe spacecraft relative motion, the relative orbit description using Keplerian orbital elements provides a better visualization of the relative motion due to the benefit of having only one term (anomaly) that changes with time out of the six orbital elements leading to the reduction of the number of terms to be tracked from six, as in the case of Hill coordinates, to one. In this paper, under certain assumptions and transformations, the spacecraft relative equations of motion, in terms of orbital element differences, is approximated into the nonlinear first kind Abel-type and Riccati-type differential equations. Furthermore, we present methodologies for the formulation of the close form analytical solutions of the approximated equations. As shown by the numerical simulations, the closed form solutions and the nonlinear equations are in conformity with Riccati-type equations having higher errors than the Abel-type equations. This shows that the Abel-type equation, a third order polynomial, approximated the relative motion better than the Riccati-type equation, a second order polynomial. The resulting new analytical solutions gave better insight into the relative motion dynamics and can be used for the analysis of spacecraft formation flying, proximity and rendezvous operations.

A Sturm\u2013Liouville Problem Arising in the Atmospheric Boundary-Layer Dynamics

We present an approach that facilitates the generation of explicit solutions to atmospheric Ekman flows with a height-dependent eddy viscosity. The approach relies on applying to the governing equations, of Sturm–Liouville type, a suitable Liouville substitution and then reducing the outcome to a nonlinear first-order differential equation of Riccati type.

A Bayesian filtering approach to layer stripping for electrical impedance tomography

Layer stripping is a method for solving inverse boundary value problems for elliptic PDEs, originally proposed in the literature for solving the Calderón problem of electrical impedance tomography (EIT), where the data consist of the Neumann-to-Dirichlet operator on the boundary. Defining a tangent–normal coordinate system near the boundary, the data are extended to a family of boundary operators on tangential surfaces inside the body, and it is shown that the operators satisfy a non-linear Riccati type differential equation with respect to the normal coordinate. The layer stripping process consists of a sequence of two alternating steps: the conductivity near the current boundary is estimated from the spatial high-frequency limit of the boundary data, and the boundary operator is propagated through a thin layer further into the domain via the Riccati equation. This way, the unknown conductivity in the interior of the domain is estimated layer by layer starting from the boundary and moving inward. The ill-posedness of the EIT problem manifests itself in such high sensitivity of the backwards Riccati equation to errors in the boundary data to cause the solutions to blow up in finite time, thus requiring regularization. In this article, we formulate the layer stripping process in the framework of Bayesian inverse problems, and we revisit the implementation in the light of Bayesian filtering. More specifically, we recast the related inverse boundary value problem as a state estimation problem, and propose an algorithm for its numerical solution based on ensemble Kalman filtering (EnKF). The new Bayesian layer stripping approach that we propose is quite robust, derivative-free and intrinsically suited for the quantification of uncertainties in the estimate. Furthermore, we show that the algorithm can be extended to realistic data collected by using a finite number of contact electrodes.

An operational method for solutions of Riccati type differential equations with functional arguments

In this article, an operational matrix approach is presented to solve the Riccati type differential equations with functional arguments. These equations are encountered in Mathematical Physics. The method is based on the least-squares approximation and the operational matrices of integration and product. By obtaining the operation matrices for each term of the problem, the method converts the problem to a system of nonlinear algebraic equations. The roots of last system are used in determination of unknown function. Error analysis is made. Numerical applications are given to show efficiency of the method and also the comparisons are made with other methods from literature. In applications of the method, it is observed from the applications that the suggested method gives effective results.

Classification of Laguerre–Hahn orthogonal polynomials of class one

AbstractWe study orthogonal polynomials related to Stieltjes functions satisfying Riccati type differential equations with polynomial coefficients, , with . We derive recurrences for the three‐term recurrence relation coefficients of the orthogonal polynomials, including connections with some forms of discrete Painlevé equations.

Affine processes beyond stochastic continuity

In this paper, we study time-inhomogeneous affine processes beyond the common assumption of stochastic continuity. In this setting, times of jumps can be both inaccessible and predictable. To this end, we develop a general theory of finite dimensional affine semimartingales under very weak assumptions. We show that the corresponding semimartingale characteristics have affine form and that the conditional characteristic function can be represented with solutions to measure differential equations of Riccati type. We prove existence of affine Markov processes and affine semimartingales under mild conditions and elaborate on examples and applications including affine processes in discrete time.

Genera of Conjoined Bases for (Non)oscillatory Linear Hamiltonian Systems: Extended Theory

In this paper we study the properties of conjoined bases of a general linear Hamiltonian system without any controllability condition. When the Legendre condition holds and the system is nonoscillatory, it is known from our previous work that conjoined bases with eventually the same image form a special structure called a genus. In this work we extend the theory of genera of conjoined bases to arbitrary systems, for which the Legendre condition is not assumed and/or the system may be oscillatory. We derive a classification of all genera of conjoined bases and show that they form a complete lattice. These results are based on the relationship between subspaces of solutions of a linear control system and orthogonal projectors satisfying a certain Riccati type differential equation. The presented theory is applied in our paper (Sepitka in Discrete Contin Dyn Syst 39(4):1685–1730, 2019) to general Riccati matrix differential equations for possibly uncontrollable linear Hamiltonian systems.

A New Algorithm for Fractional Riccati Type Differential Equations by Using Haar Wavelet

In this paper, a new collocation method based on Haar wavelet is developed for numerical solution of Riccati type differential equations with non-integer order. The fractional derivatives are considered in the Caputo sense. The method is applied to one test problem. The maximum absolute estimated error functions are calculated, and the performance of the process is demonstrated by calculating the maximum absolute estimated error functions for a distinct number of nodal points. The results show that the method is applicable and efficient.

Laguerre–Hahn orthogonal polynomials on the real line

A survey is given on sequences of orthogonal polynomials related to Stieltjes functions satisfying a Riccati type differential equation with polynomial coefficients — the so-called Laguerre–Hahn class. The main goal is to describe analytical aspects, focusing on differential equations for those orthogonal polynomials, difference and differential equations for the recurrence coefficients, and distributional equations for the corresponding linear functionals.

Third-Kind Chebyshev Wavelet Method for the Solution of Fractional Order Riccati Differential Equations

In this paper, we derive the numerical solutions of the various fractional-order Riccati type differential equations using the third-kind Chebyshev wavelet operational matrix of fractional order integration (C3WOMFI) method. The operational matrix of fractional order integration method converts the fractional differential equations to a system of algebraic equations. The third-kind Chebyshev wavelet method provides sparse coefficient matrices, therefore the computational load involved for this method is not as severe and also the resulting method is faster. The numerical solutions agree with the exact solutions for non-fractional orders, and also the solutions for the fractional orders approach those of the integer orders as the fractional order coefficient [Formula: see text] approaches to 1.

Solution of matrix differential Riccati equations in the medium of modeling system of MVTU 3.7

Purpose. For the design of systems under the action of random perturbations, a method of stochastic dynamic programming has been developed. The method is very effective because it allows one to find the optimal control of the parameters of elastic-dissipative connections of the crew suspension as a function of estimating the phase coordinates. The main reason that hinders the widespread use of this method for linear dynamic systems with many degrees of freedom is related to the lack of software for solving the matrix non-linear differential equation of Riccati type, which is included in the algorithm of the method. The aim of the work is to develop modeling schemes for solving differential equations of Riccati type in the system of modeling MVTU 3.7 (modeling in technical devices). The technique. The simulation system of MVTU 3.7 allows studying transient processes in complex dynamic systems and analyzing the stability and stability of oscillatory processes along a phase trajectory. A mathematical model of a dynamic system is created - differential equations, after - a structural diagram of the model and in the environment of MVTU 3.7 a modeling scheme is constructed. MVTU 3.7 has a graphics editor, so the modeling scheme is assembled from library blocks, which is located on the working field. The block parameters, initial conditions, integration method, simulation time, solution accuracy, etc. are set. The simulation results are given in the form of graphs. In addition, there is a text editor that records the numerical data of the simulation results. To solve differential equations like Riccati, a structural scheme is not needed. Results. A simulation scheme was created for solving differential equations of Riccati type and correlation functions of the matrix of parameters of the optimal filter were obtained. The simulation of the oscillation process of the body of the 2TE10L locomotive and the filter was performed. It has been established that for the 2TE10L locomotive it is not required to create devices for actively controlling the parameters of elastic-dissipative links. Naukova novelty. A method for solving differential equations in the simulation system of MVTU 3.7 Practical value. The main reason that prevented the widespread use of the method of stochastic dynamic programming has been eliminated and has hampered the development of the theory of creating active suspensions of transport crews.

Solution of Fractional Order Riccati Differential Equations Using Euler Wavelet Method

The fractional-order differential equations (FDEs) have the ability to model the real-life phenomena better in a variety of applied mathematics, engineering disciplines including diffusive transport, electrical networks, electromagnetic theory, probability and so forth. In most cases, there are no analytical solutions therefore a variety of numerical methods have been developed for the solution of the FDEs. In this paper, we derive the numerical solutions of the various fractional-order Riccati type differential equations using the Euler Wavelet Method (EWM). The Euler wavelet operational matrix method converts the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and efficiency of the technique.