Riccati Type Research Articles

Poincaré-Perron problem for high order differential equations in the class of almost periodic type functions

We address the Poincaré-Perron's classical problem of approximation for high order linear differential equations in the class of almost periodic type functions, extending the results for a second order linear differential equation in [23]. We obtain explicit formulae for solutions of these equations, for any fixed order n≥3, by studying a Riccati type equation associated with the logarithmic derivative of a solution. Moreover, we provide sufficient conditions to ensure the existence of a fundamental system of solutions. The fixed point Banach argument allows us to find almost periodic and asymptotically almost periodic solutions to this Riccati type equation. A decomposition property of the perturbations induces a decomposition on the Riccati type equation and its solutions. In particular, by using this decomposition we obtain asymptotically almost periodic and also p-almost periodic solutions to the Riccati type equation. We illustrate our results for a third order linear differential equation.

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.

On the solvability and the dimension of the solution set of abstract linear boundary value problems and applications to ODE'S

A solvability theory for linear abstract boundary value problems, that is of linear operator equations subject to an additional (“boundary”) condition given by a linear operator, is developed using a new approach based on normal solvability of the induced map or a pseudoinverse of the boundary map when restricted to the null space of the homogeneous problem. Uniqueness and dimension of their solution sets is also established. Applications to BVP's for finite-dimensional as well as infinite-dimensional systems of ordinary differential equations defined on the half line or the whole line are given. Systems having exponential dichotomy, or satisfying Riccati type inequalities are studied, as well as systems with asymptotically hyperbolic linear part and systems between pairs of admissible Banach spaces.

Shifted Legendre method for solutions of Riccati type functional differential equations and application of RL circuit model

AbstractIn order to solve the Riccati type functional differential equations under mixed condition, this research suggests a collocation approach based on shifted Legendre polynomials. These equations are seen in physics and electronic sciences. By means of the evenly spaced collocation points, the shifted Legendre polynomials, their derivatives and matrix forms, the method is constructed. It reduces the problem into a system nonlinear algebraic equations. Error analysis is made. The method is applied to numerical examples and their results are compared with other methods from literature. And also, the method is applied to nonlinear RL electrical circuit model. The results show that the method is effective.

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.

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.

On the stochastic linear quadratic optimal control problem by piecewise constant controls: The infinite horizon time case

This paper is devoted to the problem of indefinite stochastic linear quadratic (LQ) optimal control by piecewise constant controls in an infinite horizon case. By restricting the set of admissible controls to the class of piecewise constant stochastic processes, we reformulated the above control problem under the setting of systems modeled by Itô differential equations controlled by impulses. We show that the solution in a state feedback form of the indefinite stochastic LQ control problem is equivalent to the existence of a global stabilizing solution associated to a class of backward matrix linear differential equations with a Riccati type jumping operators.

Towards a Holographic-Type Perspective in the Analysis of Complex-System Dynamics

By operating with the Scale Relativity Theory by means of two scenarios (Schrӧdinger and Madelung-type scenarios) in the dynamics of complex systems, we can achieve a description of these complex systems through a holographic-type perspective. Then, a gauge invariance of the Riccati type becomes functional in complex-system dynamics, which implies several consequences: conservation laws (in particular, for dynamics, the kinetic momentum conservation law), simultaneity and synchronization among the structural units’ (belonging to a complex system) dynamics, and temporal patterns through harmonic mappings. Finally, an economic case analysis is highlighted.

The forbidden set, solvability and stability of a circular system of complex Riccati type difference equations

&lt;abstract&gt;&lt;p&gt;In this paper, the circular system of Riccati type complex difference equations of the form&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ u_{n+1}^{(j)} = \frac{a_ju_n^{(j-1)}+b_j}{c_ju_n^{(j-1)}+d_j}, \; n = 0, 1, 2, \cdots, \; j = 1, 2, \cdots, k, $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;where $ u_n^{(0)}: = u_n^{(k)} $ for all $ n $, is investigated. First, the forbidden set of the equation is given. Then the solvability of the system is examined and the expression of the solutions, given in terms of their initial values. Next, the asymptotic behaviour of the solutions is studied. Finally, in case of negative Riccati real numbers&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE2"&gt; \begin{document}$ R_j: = \frac{a_jd_j-b_jc_j}{[a_j+d_j]^2}, \; j\in\overline{1, k}, $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;it is shown that there exists a unique positive fixed point which attracts all solutions starting from positive states.&lt;/p&gt;&lt;/abstract&gt;

NON-LINEAR BEHAVIORS IN THE DYNAMICS OF COMPLEX SYSTEMS WITH POTENTIAL ECONOMY APPLICATION. QUALITATIVE ANALYSIS FROM MULTI-FRACTAL PERSPECTIVE

"In a Schrödinger-type and Madelung-type scenarios for the description of complex economics system dynamics, SL(2R) symmetries are highlighted. The emergence of such symmetries has several consequences: the existence of analogic-type behavior as a gauge invariance of Riccati type as well as the existence of digital-type behavior through the spontaneous symmetry breaking of the same gauge invariance. When said symmetries are discussed in the context of economics dynamics, the individual reaction to market signals can be associated to period doubling and modulated dynamics (i.e. to the digital signals) while, the behaviors of large investors and of the State, through banking or monetary policies, can associated to the “complex economics system background” (i.e. analogical signals). Moreover, the markets have a fractal/multi-fractal structure on the long term, being characterized by a “self-memory”. The economic structures emphasize fluctuations but, they never reach the chaos state. Thus, a holographic approach on complex economics system dynamics (and, on economics complex economics systems) provides a valid and more natural perspective, compared to the standard approaches. Our research provides a qualitative insight of economics complex system dynamics, remaining a more rigorous study which reveals a quantitative analysis of financial fractal bubbles to be done in further research."

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.

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.

Planning the Number of Aircraft in a Group Flight with their Survivability and the Required Observation Duration of Ground Objects

In the interests of improving the quality of a group aircraft flight planning, the developing algorithms problem formulation for the operational determination of the permissible observation duration for ground objects is formulated. An algorithm for determining the observation duration when servicing the next request to be described in the form of a fuzzy logic procedure is proposed. To implement the algorithm for determining the duration of observation, a specialized expert system has been developed. The input of the expert system receives values that describe the influence of the factors in assessing the priority of servicing the next object. At the output of the expert system, an alternative is formed to continue searching for an object or to stop. A new approach to solving the target distribution problem between aircraft in a group flight is proposed. It is based on the joint use of two dynamic priorities in the minimax algorithm for selecting observation objects and for assigning service aircraft. An original approach to determine the rational aircrafts composition in one flight is proposed. This is done with the help of the queuing theory apparatus, which takes into account the random nature of the dynamic situation. To estimate the required aircrafts composition when servicing the flow of requests, the countermeasures process is described using two nonlinear differential equations (of the Riccati type). A general formula for determining aircrafts composition in one flight has been obtained. Mathematical model of the aircraft survivability loss in the form of the Bernoulli equation is formed. Computer simulation of the aircraft survivability losses in one flight was carried out for three cases: with weak interference, with equal opposing forces, with strong counteraction.

Drug-Loaded Polymeric Particulated Systems for Ophthalmic Drugs Release.

Drug delivery to the anterior or posterior segments of the eye is a major challenge due to the protection barriers and removal mechanisms associated with the unique anatomical and physiological nature of the ocular system. The paper presents the preparation and characterization of drug-loaded polymeric particulated systems based on pre-emulsion coated with biodegradable polymers. Low molecular weight biopolymers (chitosan, sodium hyaluronate and heparin sodium) were selected due to their ability to attach polymer chains to the surface of the growing system. The particulated systems with dimensions of 190–270 nm and a zeta potential varying from −37 mV to +24 mV depending on the biopolymer charges have been obtained. Current studies show that particles release drugs (dexamethasone/pilocarpine/bevacizumab) in a safe and effective manner, maintaining therapeutic concentration for a longer period of time. An extensive modeling study was performed in order to evaluate the drug release profile from the prepared systems. In a multifractal paradigm of motion, nonlinear behaviors of a drug delivery system are analyzed in the fractal theory of motion, in order to correlate the drug structure with polymer. Then, the functionality of a SL(2R) type “hidden symmetry” implies, through a Riccati type gauge, different “synchronization modes” (period doubling, damped oscillations, quasi-periodicity and intermittency) during the drug release process. Among these, a special mode of Kink type, better reflects the empirical data. The fractal study indicated more complex interactions between the angiogenesis inhibitor Bevacizumab and polymeric structure.

New criteria for oscillation of damped fractional partial differential equations

&lt;abstract&gt;&lt;p&gt;In this paper, we consider a class of fractional partial differential equations with damping term subject to Robin and Dirichlet boundary value conditions. We derive several new sufficient criteria for oscillation of solutions of the equations by using the integral averaging technique and generalized Riccati type transformations. Some applications of the main results are illustrated by some examples.&lt;/p&gt;&lt;/abstract&gt;

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.

Solving Riccati Type q−Difference Equations via Difference Transform Method

In this paper, we deal on the time scale that its delta derivative of graininess function is a nonzero positive constant. Based on the Taylor formula for this time scale, we investigate the difference transform method (DTM). This method has been applied successfully to solve Riccati type difference equations in quantum calculus. To demonstrate the ability and efficacy of this method, some examples have been provided.

Uniqueness of Solutions for the Initial Value Problem of a Simple Type Riccati Equation with Variable Fractional Order

In recent years, researches on the fractional Riccati equation have attracted extensive attention, and many achievements have emerged. However, most of the current works are carried out on the basis of constant fractional order. Here we study the Cauchy problem of one simple type Riccati equation with variable fractional order. By using Gronwall-Bellman inequality and other mathematical tools, we will prove the uniqueness of solutions of this problem.

Distributed filtering stochastic parameter learning algorithm for discrete-time random nonlinear systems with delay

Distributed filtering algorithm for a discrete time random nonlinear stochastic systems associated with state delay for the distributed wireless communication sensors are discussed in this paper. Stochastic Parameter Learning Algorithm (SPLA) is tend to aim at obtaining a collection of stochastic filter parameters in a finite limited time horizon, that minimize the traces of the upper limits which is permitted to reduce the error variance matrices of the concerned stochastic filter system’s states and delay measurements. Filter gain values of the filter derived by the determination of Riccati type difference equations, estimates systems states with delay. Two different filter rules are taken into account for the SPLA discrete time random nonlinear systems with steady state space equations model. Zero mean distinct covariance matrix along with the constructive state values and constant time delay are focused in compatible dimensions. The variance of the projected systems predicted noise and the actual estimated noise are validated through numerical examples.