Nonlinear Matrix Differential Equations Research Articles

On the Ψ − Conditional Asymptotic Stability of a Nonlinear Lyapunov Matrix Differential Equation with Integral Term as Right Side

Abstract The aim of this paper is to give sufficient conditions for Ψ − conditional asymptotic stability of the Ψ − bounded solutions of a nonlinear Lyapunov matrix differential equation with integral term as right side.

Stability and Ultimate Boundedness of Solutions of Certain Third Order Nonlinear Rectangular Matrix Differential Equations

We present in this paper the qualitative study of solutions of certain third order nonlinear matrix differential equations where the unknown function X is matrix-valued. The properties of solutions were investigated using Lyapunov’s direct method by employing the use of suitable Lyapunov functionals obtained from the differential equations describing the system satisfying certain requirements for establishing the stability and boundedness of solutions of the system considered. An example is given to demonstrate the significance of the results obtained as well as analysis through geometric graphs describing the dynamics of the system’s solutions. The results obtained are novel and will significantly enhance and extend the results of those mentioned in the literature.

Elzaki transform decomposition approach to solve Riccati matrix differential equations

Elzaki Transform Adomian decomposition technique (ETADM), which an elegant combine, has been employed in this work to solve non-linear Riccati matrix differential equations. Solutions are presented to demonstrate the relevance of the current approach. With the use of figures, the results of the proposed strategy are displayed and evaluated. It is demonstrated that the suggested approach is effective, dependable, and simple to apply to a range of related scientific and technical problems.

Solving nonlinear matrix and Riesz-Caputo fractional differential equations via fixed point theory in partial metric spaces

A modified implicit relation and ?-implicit contractive condition are introduced in the setting of relational partial metric spaces and some related fixed point results are derived. Two suitable examples are provided. As an application, sufficient conditions are derived for the existence of a unique positive definite solution of the non-linear matrix equation X = B + Pk i=1A* i T(X)Ai. An example is given, using matrices that are randomly generated, as well as convergence and error analysis and average CPU time analysis. Solving fractional differential equations of Riesz-Caputo type with anti-periodic boundary conditions is also discussed, followed by two illustrations.

On the Ψ − Conditional Exponential Asymptotic Stability of a Nonlinear Lyapunov Matrix Differential Equation with Integral Term as Right Side

Abstract The aim of this paper is to give sufficient conditions for Ψ − conditional exponential asymptotic stability of the Ψ − bounded solutions of a nonlinear Lyapunov matrix differential equation with integral term as right side.

On the Ψ−uniform asymptotic stability of nonlinear Lyapunov matrix differential equations

This paper deals with obtaining (necessary and) sufficient conditions for Ψ− uniform asymptotic stability of solutions of nonlinear Lyapunov matrix differential equations.

On the Approximated Solution of a Special Type of Nonlinear Third-Order Matrix Ordinary Differential Problem

Matrix differential equations are at the heart of many science and engineering problems. In this paper, a procedure based on higher-order matrix splines is proposed to provide the approximated numerical solution of special nonlinear third-order matrix differential equations, having the form Y(3)(x)=f(x,Y(x)). Some numerical test problems are also included, whose solutions are computed by our method.

On the $Psi$-instability of a nonlinear Lyapunov matrix differential equation with integral term as right side

The aim of this paper is to give sufficient conditions for $Psi$-instability of trivial solution of a nonlinear Lyapunov matrix differential equation with integral term as right side.

On the bounded and stabilizing solution of a generalized Riccati differential equation arising in connection with a zero‐sum linear quadratic stochastic differential game

SummaryWe study a class of coupled nonlinear matrix differential equations arising in connection with the solution of a zero‐sum two‐player linear quadratic (LQ) differential game for a dynamical system modeled by an Itô differential equation subject to random switching of its coefficients. The system of differential equations under consideration contains as special cases the game‐theoretic Riccati differential equations arising in the solution of the H∞ control problem from the deterministic and stochastic cases. A set of sufficient conditions that guarantee the existence of the bounded and stabilizing solution of this kind of Riccati differential equations is provided. We show how such stabilizing solution is involved in the construction of the equilibrium strategy of a zero‐sum LQ stochastic differential game on an infinite‐time horizon and give as a byproduct the solution of such a control problem.

Moore-Penrose Generalized Inverse to Kronecker Product Matrix Boundary Value Problems

This paper deals with the existence-uniqueness to Kronecker product (KP) 3- point boundary value problems(TBVP) for the first order linear system as well as nonlinear matrix differential equations with the help of Moore-Penrose generalized inverse and Banach fixed point theory.

Design of Terminal Sliding Mode Controllers for Disturbed Non-Linear Systems Described by Matrix Differential Equations of the Second and First Orders

This paper describes a design scheme for terminal sliding mode controllers of certain types of non-linear dynamical systems. Two classes of such systems are considered: the dynamic behavior of the first class of systems is described by non-linear second-order matrix differential equations, and the other class is described by non-linear first-order matrix differential equations. These two classes of non-linear systems are not completely disjointed, and are, therefore, investigated together; however, they are certainly not equivalent. In both cases, the systems experience unknown disturbances which are considered bounded. Sliding surfaces are defined by equations combining the state of the system and the expected trajectory. The control laws are drawn to force the system trajectory from an initial condition to the defined sliding surface in finite time. After reaching the sliding surface, the system trajectory remains on it. The effectiveness of the approaches proposed is verified by a few computer simulation examples.

An explicit Floquet-type representation of Riccati aperiodic exponential semigroups

The article presents a rather surprising Floquet-type representation of time-varying transition matrices associated with a class of nonlinear matrix differential Riccati equations. The main difference with conventional Floquet theory comes from the fact that the underlying flow of the solution matrix is aperiodic. The monodromy matrix associated with this Floquet representation coincides with the exponential (fundamental) matrix associated with the stabilising fixed point of the Riccati equation. The second part of this article is dedicated to the application of this representation to the stability of matrix differential Riccati equations. We provide refined global and local contraction inequalities for the Riccati exponential semigroup that depend linearly on the spectral norm of the initial condition. These refinements improve upon existing results and are a direct consequence of the Floquet-type representation, yielding what seems to be the first results of this type for this class of models.

On ψ Bounded Solutions for a Nonlinear Lyapunov Matrix Differential Equation on 𝕉

Abstract Using Banach and Schauder - Tychono fixed point theorems, existence results for a nonlinear Lyapunov matrix differential equation on 𝕉 are given. The obtained results generalize and extend the results from [5] and [18].

On the Ψ-Conditional Exponential Asymptotic Stability of Nonlinear Lyapunov Matrix Differential Equations

Abstract It is proved (necessary and) sufficient conditions for Ψ– conditional exponential asymptotic stability of the trivial solution of nonlinear Lyapunov matrix differential equations

Optimal control of two coupled spinning particles in the Euler–Lagrange picture

A family of optimal control problems for a single and two coupled spinning particles in the Euler–Lagrange formalism is discussed. A characteristic of such problems is that the equations controlling the system are implicit and a reduction procedure to deal with them must be carried out. The reduction of the implicit control equations arising in these problems will be discussed in the slightly more general setting of implicit equations defined by invariant one-forms on Lie groups. As an example the first order differential equations describing the extremal solutions of an optimal control problem for a single spinning particle, obtained by using Pontryagin’s Maximum Principle (PMP), will be found and shown to be completely integrable. Then, again using PMP, solutions for the problem of two coupled spinning particles will be characterized as solutions of a system of coupled non-linear matrix differential equations. The reduction of the implicit system will show that the reduced space for them is the product of the space of states for the independent systems, implying the absence of ‘entanglement’ in this instance. Finally, it will be shown that, in the case of identical systems, the degree three matrix polynomial differential equations determined by the optimal feedback law, constitute a completely integrable Hamiltonian system and some of its solutions are described explicitly.

On the Ψ-Conditional Asymptotic Stability of Nonlinear Lyapunov Matrix Differential Equations

Abstract It is proved (necessary and) sufficient conditions for Ψ − conditional asymptotic stability of the trivial solution of linear or nonlinear Lyapunov matrix differential equations.

On the Ψ-Strong Stability of Nonlinear Lyapunov Matrix Differential Equations

Abstract The paper provides (necessary and) sufficient conditions for Ψ-strong stability of the trivial solution of a linear Lyapunov matrix differential equations. Further, sufficient condition are obtained for Ψ-strong stability of the trivial solution of a nonlinear Lyapunov matrix differential equation.

On the Ψ – Conditional Stability of Nonlinear Lyapunov Matrix Differential Equations

AbstractIt is proved (necessary and) suficient conditions for Ψ conditional stability of the trivial solution of linear or nonlinear Lyapunov matrix differential equations

A Discrete-Time Algorithm for the Resolution of the Nonlinear Riccati Matrix Differential Equation for the Optimal Control

The Riccati Matrix Differential Equation (RMDE) is an interesting equation in different fields of science and engineering practice. In fact, that the arithmetic solution for this matrix differential equation in the general case of varying-time matrices is very difficult to find. The literature offers the solution of this differential equation in the case of dependant constant matrices (i.e. invariant-time matrices). The present approach is an approximate discrete-time method for the resolution of the matrix differential equation of Riccati in the general case of varying-time (dependant of time) matrices; the method in fact, is a discrétisation of the exact matrix solution, that evaluates for any so small step of time, and which is function of the solution of the preceding step of time and the constitute equation matrices. The proposed algorithm is verified, for a controlled structure under Modified El-Centro earthquake by a comparison with the same uncontrolled structure, which constitutes by a two Degrees Of Freedom (2DOF) system. The results of this comparison offer good differences between the controlled and the uncontrolled systems.

Gain-scheduled controllers design for interceptor parameter-varying system with multi-saturated constraint

Based on homogeneous polynomial parameter dependent Lyapunov function (HPPDLF) theory and the algorithm that the nonlinear matrix differential equations converted into convex polyhedron state equations, the robust optimal control system is designed, for the interceptor with time-varying parameters and actuator nonlinear multi-saturated constraint. First, the model of the actuator saturation operator is established, which provides the control system topology. The nonlinear differential equations with time-varying parameters converted into the linear polytopic vertex system by introduced the normalization factors of the amplitude and rate actuator saturation, and then it converted into HPPD matrix differential equations, based on HPPDLF theory and the extension of Pólya’s theorem to the case of matrix-valued polynomials. The dynamic performance of the closed-loop control system is obtained by setting the minimum allowable value of the scaling factor, when solving the minimum of the maximum homothetic set in the iterative processing. Finally the simulations of the interceptor have demonstrated the effectiveness of the approach proposed.