Riccati Solution Research Articles

Extendability and qualitative properties of solutions of Riccati's equation

На вещественной оси рассматривается уравнение Риккати с непрерывными коэффициентами и неотрицательным дискриминантом правой части. Исследуется продолжаемость его решений на бесконечный интервал. Найдена асимптотика его решений в зависимости от их начальных значений и свойств функций, являющихся корнями правой части уравнения. Получены результаты об асимптотическом поведении решений, определeнных в окрестности $\pm\infty$. Исследована структура множества ограниченных решений уравнения в случае, когда корни правой части уравнения - различные на всей их области определения непрерывно дифференцируемые функции, монотонно стремящиеся к пределам при $x\to\pm\infty$. Дополнены, усилены или уточнены некоторые известные результаты. Библиография: 47 названий.

On special solutions to the Ermakov–Painlevé XXV equation

In this paper, we study a nonlinear second-order ordinary differential equation which we call the Ermakov–Painlevé XXV equation since under certain restrictions on its coefficients it can be reduced either to the Ermakov or the Painlevé XXV equation. The Ermakov–Painlevé XXV equation arises from a generalized Riccati equation and the related third-order linear differential equation via the Schwarzian derivative. The generalized Riccati equation has two families of Riccati solutions and we study the corresponding solutions to the Ermakov–Painlevé XXV equation. We show that one of these families appears only in the Ermakov case. We give examples of the Ermakov–Painlevé XXV equations and show how to construct their solutions expressed in terms of elementary or in terms of the classical special functions.

Exact learning dynamics of deep linear networks with prior knowledge *

Learning in deep neural networks is known to depend critically on the knowledge embedded in the initial network weights. However, few theoretical results have precisely linked prior knowledge to learning dynamics. Here we derive exact solutions to the dynamics of learning with rich prior knowledge in deep linear networks by generalising Fukumizu’s matrix Riccati solution (Fukumizu 1998 Gen 1 1E–03). We obtain explicit expressions for the evolving network function, hidden representational similarity, and neural tangent kernel over training for a broad class of initialisations and tasks. The expressions reveal a class of task-independent initialisations that radically alter learning dynamics from slow non-linear dynamics to fast exponential trajectories while converging to a global optimum with identical representational similarity, dissociating learning trajectories from the structure of initial internal representations. We characterise how network weights dynamically align with task structure, rigorously justifying why previous solutions successfully described learning from small initial weights without incorporating their fine-scale structure. Finally, we discuss the implications of these findings for continual learning, reversal learning and learning of structured knowledge. Taken together, our results provide a mathematical toolkit for understanding the impact of prior knowledge on deep learning.

Retrieval of Optical Solitons with Anti-Cubic Nonlinearity

Purpose: In this article, two main subjects are discussed. First, the nonlinear Schrödinger equation (NLSE) with an anti-cubic (AC) nonlinearity equation is examined, which has a great working area, importance and popularity among the study areas of soliton behavior in optical fibers, by using the enhanced modified extended tanh expansion method and a wide range of optical soliton solutions is obtained. Second, the effects of AC parameters on soliton behavior are examined for each obtained soliton type. Methodology: In order to apply the method, the non-linear ordinary differential equation form (NLODE) of the investigated NLSE-AC is obtained by applying the defined wave transformation. Then, with the help of the proposed algorithm for the NLODE form, polynomial form, an algebraic equation system is obtained by setting the coefficients of this form to zero, and the solution of this system is also obtained. After determining the suitable solution set, the optical soliton solution of the investigated problem is obtained with the help of the serial form of the proposed method, a Riccati solution and wave transform. After checking that the solution satisfies the investigated problem, 3D and 2D graphics are obtained for the special parameter values and the necessary comments are made in the relevant sections. Findings: With the proposed method, many optical soliton solutions, such as topological, anti-peaked, combined peaked-bright, combined anti-peaked dark, singular, combined singular-anti peaked, periodic singular, composite kink anti-peaked, kink, periodic and periodic, with different amplitudes are obtained, and 3D and 2D representations have been made. Then, the effect of AC parameters on the soliton behavior in each case has been successfully studied. It has been shown that AC parameters have a significant effect on the soliton behavior, and this effect changes depending on the soliton shape and the parameters. Moreover, providing and maintaining the delicate balance between the soliton shape and the parameters and the interaction of the parameters with each other involves great difficulties. Originality: Although some soliton types of the NLSE-AC equation have been presented for the first time in this study, there is no study in the literature showing the effect of AC parameters on soliton behavior, especially for the NLSE-AC equation.

Utility Maximization in Multivariate Volterra Models

This paper is concerned with portfolio selection for an investor with power utility in multiasset financial markets in a rough stochastic environment. We investigate Merton’s portfolio problem for different multivariate Volterra models, covering the rough Heston model. First we consider a class of multivariate affine Volterra models introduced in [E. Abi Jaber, E. Miller, and H. Pham, SIAM J. Financial Math.., 12 (2021), pp. 369–409]. Based on the classical Wishart model described in [N. Bäuerle and Z. Li, J. Appl. Probab., 50 (2013), pp. 1025–1043], we then introduce a new matrix-valued stochastic volatility model, where the volatility is driven by a Volterra–Wishart process. Due to the non-Markovianity of the underlying processes, the classical stochastic control approach cannot be applied in these settings. To overcome this issue, we provide a verification argument using calculus of convolutions and resolvents. The resulting optimal strategy can then be expressed explicitly in terms of the solution of a multivariate Riccati–Volterra equation. We thus extend the results obtained by Han and Wong to the multivariate case, avoiding restrictions on the correlation structure linked to the martingale distortion transformation used in [B. Han and H. Y. Wong, Finance Res. Lett., 39 (2021)]. We also provide existence and uniqueness theorems for the occurring Volterra processes and illustrate our results with a numerical study.

Bright, dark and hybrid multistrip optical soliton solutions of a non-linear Schrödinger equation using modified extended tanh technique with new Riccati solutions

Bright, dark and hybrid multistrip optical soliton solutions of a non-linear Schrödinger equation using modified extended tanh technique with new Riccati solutions

The Haar wavelets based numerical solution of Reccati equation with integral boundary condition

A Haar wavelet collocation method (HWCM) is presented for the solution of Riccati equation subject to the two-point and integral boundary condition. The qua?silinearization technique is applied to linearized the Riccati equation and then the linearized equation with boundary condition is solved by converting into system of algebraic equation with the help of Haar wavelets. We have considered three different form of Reccati equation, two having integral boundary condition and one with two-point boundary condition. The numerical results obtained by HWCM are stable, efficient and convergent.

Fast iterative solutions of Riccati and Lyapunov equations

AbstractIn this article, new iterative algorithms for solving the discrete Riccati and Lyapunov equations are derived in the case where the transition matrix is diagonalizable with real eigenvalues. It is shown that the proposed iterative algorithms are faster than the classical ones, even for a small number of iterations.

Fully probabilistic control for uncertain nonlinear stochastic systems

AbstractThis paper develops a novel probabilistic framework for stochastic nonlinear and uncertain control problems. The proposed framework exploits the Kullback–Leibler divergence to measure the divergence between the distribution of the closed‐loop behavior of a dynamical system and a predefined ideal distribution. To facilitate the derivation of the analytic solution of the randomized controllers for nonlinear systems, transformation methods are applied such that the dynamics of the controlled system becomes affine in the state and control input. Additionally, knowledge of uncertainty is taken into consideration in the derivation of the randomized controller. The derived analytic solution of the randomized controller is shown to be obtained from a generalized state‐dependent Riccati solution that takes into consideration the state‐ and control‐dependent functional uncertainty of the controlled system. The proposed framework is demonstrated on an inverted pendulum on a cart problem, and the results are obtained.

Solving the nonlinear Riccati equation for the 1D plane-wave reflection response from a velocity gradient interface by a Heun to hypergeometric reduction

For 1D acoustic wave propagation in the frequency domain, the complex nonlinear Riccati differential equation describes the dependence of the plane-wave reflection response in depth, defined as the upgoing part of the wavefield divided by the downgoing part of the wavefield at that depth. In the case that the model is a velocity gradient interface expressed in terms of a smooth Heaviside function defined by the Fermi-Dirac function, the Riccati equation is shown to have an analytical solution for the reflection response. The solution accounts for all wave phenomena in the gradient zone. The solution of the Riccati equation for the reflection response is obtained by introducing the Riccati equation for the reciprocal of the reflection response which can be related to a second-order self-adjoint linear differential equation. The two latter equations obey the same radiation condition. Furthermore, a connection between the second-order linear equation and the Heun equation is obtained, both being characterized by having four regular singular points. By a change of variables, the Heun equation can be transformed into Gauss’ hypergeometric equation, whereby the solution is expressed as the sum of two linearly independent functions containing hypergeometric functions. The advantage of applying the Heun-to-hypergeometric reduction to the nonlinear Riccati equation is that solutions to the hypergeometric equation are well known. The two functions allow us to analytically determine the Riccati solution because the ratio of the linearly independent functions at infinity is constant. In contrast to the classic frequency-independent reflection coefficient from two layers in welded contact, the reflection coefficient at the surface associated with the gradient interface depends on frequency. In the limit of zero frequency, the reflection coefficient approaches the classic one. We have implemented the iterative solution of the integral form of the Riccati equation and found that the solution builds up nonlinearly to the correct solution. The model is verified by a numerical example valid for 1D wave propagation.

Approximate solutions for the fractional order quadratic Riccati and Bagley-Torvik differential equations

The Galerkin method is presented and applied for getting semi-analytical solutions of quadratic Riccati and Bagley-Torvik differential equations in fractional order. New theorems are proved to minimize the generated residual after invoking the Legendre polynomials as a basis in the Galerkin method. The proposed method is compared with other methods by solving some initial value problems of different fractional orders. The comparisons and results are illustrated via tables and figures. It can be concluded that the Legendre-Galerkin method is convenient for these problems due to its efficiency and reliability.

The $\mathcal{H}_2$-optimal Control Problem of CSVIU Systems: Discounted, Counterdiscounted, and Long-Run Solutions

The paper deals with stochastic control problems associated with $H_2$ performance indices such as energy or power norms or energy measurements. The control applies to a class of systems for which a stochastic process conveys the underlying uncertainties, known as control and state variation increase uncertainty (CSVIU). These indices allow various emphases, focusing on the transient behavior with the discounted norm to stricter conditions on stability and steady-state mean-square error and convergence rate, using the optimal overtaking criterion. The long-run average power control stands as a midpoint in this respect. A critical advance regards the explicit form of the optimal control law, expressed in two forms. One takes a perturbed affine Riccati-like form of feedback solution; the other comes from a generalized normal equation that arises from the nondifferentiable local optimal problem. They are equivalent, but the former allows optimality and stability, and the latter develops into a search method to attain the optimal law. A detectability notion and a Riccati solution grant stochastic stability from the behavior of the norms. The energy overtaking criterion requires a further constraint on a matrix spectral radius. With these findings, the paper revisits the rising of the inaction solution, a prominent feature of CSVIU models to deal with the uncertainty inherent to poorly known models. Besides the optimal solution, it provides the tools to pursue it.

Inhomogeneous affine Volterra processes

We extend recent results on affine Volterra processes to the inhomogeneous case. This includes moment bounds of solutions of Volterra equations driven by a Brownian motion with an inhomogeneous kernel K(t,s) and inhomogeneous drift and diffusion coefficients b(s,Xs) and σ(s,Xs). In the case of affine b and σσT we show how the conditional Fourier–Laplace functional can be represented by a solution of an inhomogeneous Riccati–Volterra integral equation. For a kernel of convolution type K(t,s)=K¯(t−s) we establish existence of a solution to the stochastic inhomogeneous Volterra equation. If in addition b and σσT are affine, we prove that the conditional Fourier–Laplace functional is exponential–affine in the past path. Finally, we apply these results to an inhomogeneous extension of the rough Heston model used in mathematical finance.

Differential equations for the recurrence coefficients of semiclassical orthogonal polynomials and their relation to the Painlevé equations via the geometric approach

AbstractIn this paper, we present a general scheme for how to relate differential equations for the recurrence coefficients of semiclassical orthogonal polynomials to the Painlevé equations using the geometric framework of the Okamoto Space of Initial Conditions. We demonstrate this procedure in two examples. For semiclassical Laguerre polynomials appearing in Ref. 1, we show how the recurrence coefficients are connected to the fourth Painlevé equation. For discrete orthogonal polynomials associated with the hypergeometric weight appearing in Ref. 2, we discuss the relation of the recurrence coefficients to the sixth Painlevé equation, extending the results of Ref. 3, where a similar approach was used for a discrete system for the same recurrence coefficients. Though the discrete and differential systems here share the same geometry, the construction of the space of initial conditions from the differential system is different and reveals extra considerations that must be made. We also discuss a number of related topics in the context of the geometric approach, such as Hamiltonian forms of the differential equations for the recurrence coefficients, Riccati solutions for special parameter values, and associated discrete Painlevé equations.

A fundamental solution concept for a family of state constrained regulator problems

A family of continuous-time regulator problems, subject to linear time-varying dynamics and convex state constraints, is parameterized with respect to a space of semiconcave terminal costs. By embedding the state constraints into an extended real-valued running cost, a family of relaxations is employed to develop a min-plus convolution-based fundamental solution concept for the family of regulator problems. A representation of the fundamental solution concept is provided in terms of solutions of associated differential Riccati and related equations, and possible avenues for efficient numerical computation are explored.

On the optical soliton solution of the [formula omitted] dimensional perturbed NLSE in optical nano-fibers

This study investigates new exact solutions to a perturbed non-linear Schrödinger equation (PNLSE). We have successfully obtained lots of solutions to the considered equations such as singular, dark, singular periodic waves, combined dark-singular soliton, and rational function solutions using the modified extended tanh expansion method (METEM) enhanced with new Riccati solutions (NRCS). The contour, 2D, and 3D plots of the obtained solutions are depicted using Matlab and all other computations carried out by Maple. It is proven that the method is so powerful and effective in solving many non-linear science problems.

A semi-analytical solutions of fractional Riccati's differential equation via singular and non-singular operators

Looking for offer a solution of the Riccati's differential equations of fractional order (FRDEs) involving Caputo derivative (CD), Caputo-Fabrizio derivative (CFD) or Atangana-Baleanu derivative (ABD) in this comparative research is based on a semi-analytical iterative approach. Temimi and Ansari introduced this method and called it TAM. The comparison of the time used in minutes is given for three derivatives CD, CFD and ABD. Meanwhile, the comparison of the approximate solutions with CD, CFD and ABD are presented. Regarding the help of the software Mathematica, all the results have been obtained and the calculations have been done.

An approximate solution of Riccati’s differential equation using fuzzy linguistic model

Looking for offer, an approximate solution of the Riccati’s differential equations in this research work is based on a fuzzy linguistic model (FLM). It is shown by the FLM that it is possible to approximate continuous functions which are satisfied at the interpolation points. To obtain the function as the approximation of the solution of the RDEs, fuzzy linguistic model (FLM) was used. In numerical approaches, in order to find an approximation for a function that exists on a particular interval, just a limited number of points are utilized. However, the distinctive feature of the FLM in comparison with other approaches is that in this method all points in the arbitrary interval are used. Five examples along with their approximate solutions have been presented to show the suitability, convenience, simplicity and performance of the method. The outcomes demonstrate that the advantage of this approach compared to other methods is that there are no limiting conditions for nonlinear differential equations of order one or two with initial conditions or boundary conditions.

On Expansions of Solutions of Riccati's Equation in Asymptotic Series

Рассматриваются скалярные вещественные уравнения Риккати с коэффициентами, раскладывающимися в сходящиеся степенные ряды в окрестности бесконечности. Исследуются продолжаемые решения таких уравнений. Методами степенной геометрии получены условия разложений этих решений в асимптотические ряды. Библиография: 8 названий.

On the Semi-Global Stability of an EK-Like Filter

This letter proposes to apply the Kalman-like observer paradigm to general nonlinear systems by linearization along the estimated trajectory, similarly to an Extended Kalman Filter. The main difference is that the quadratic Riccati equation is replaced by a linear Lyapunov equation which can be solved and explicitly related to a determinability Gramian. This allows to show by Lyapunov analysis and without any ad-hoc assumption on the Riccati solution, that the resulting observer, called Extended Kalman-like Filter, can be made semi-globally convergent if the input is actively used to a) stabilize the (unknown) true trajectory, b) sufficiently excite the determinability of the linearized systems along the (known) estimated trajectory. A class of systems where this compromise can be reached is provided.