Riccati Equation Research Articles

LQ control of system governed by backward stochastic difference equations and applications

This paper investigates the linear quadratic (LQ) optimal control of system governed by backward stochastic difference equations. In particular, multiple multiplicative noises are involved in the stochastic system, which leads to the situation that the controller uses partial information to complete optimisation. The main contribution is to give the explicitly optimal controller based on introduced Riccati equation. The key technique is to solve the corresponding forward–backward stochastic difference equations by establishing the non-homogeneous relationship, which is completely different from the homogeneous relationship in standard optimal control of system governed by forward stochastic difference equations. Moreover, the derived results are applied to stochastic optimal control problem with terminal constraints as applications.

The time-dependent harmonic oscillator revisited

Abstract We point out a rather effective approach for solving the time-dependent harmonic oscillator $\ddot q=-\omega^2 q$ under various regularity assumptions. Where $\omega(t)$ is $C^1$ this is reduced to Hamilton equation for the angle variable $\psi$ alone (the action variable $I$ is obtained by quadrature). The fixed point theorem for the integral equation equivalent to the generic Cauchy problem for $\psi(t)$ 
yields a sequence $\{\psi^{(h)}\}_{h\in \mathbb{N}_0}$ converging to $\psi$ rather fast; if $\omega$ varies slowly or little, already $\psi^{(0)}$ approximates $\psi$ well for rather long time lapses. The discontinuities of $\omega$, if any, determine those of $\psi,I$. 
The zeroes of $q,\dot q$ are investigated via Riccati equations. Our approach may simplify the study of: upper and lower bounds 
on the solutions; the stability of the trivial one; parametric resonance when $\omega(t)$ is periodic; the adiabatic invariance of $I$; asymptotic expansions in a slow time parameter $\varepsilon$; time-dependent driven and damped parametric oscillators; etc.

Exploring the impact of Brownian motion on novel closed-form solutions of the extended Kairat-II equation.

This work considers a stochastic form of an extended version of the Kairat-II equation by adding Browning motion into the deterministic equation. Two analytical approaches are utilized to derive analytical solutions of the modified equation. The first method is the modified Tanh technique linked with the Riccati equation, which is implemented to extract some closed-form solutions in the form of tangent and cotangent functions. The second technique is the Sardar sub-equation method (SSEM) which is used to attain several analytical solutions in the form of trigonometric and hyperbolic functions. Solutions selected randomly from the large families of solutions with suggested techniques are visualized in 3D and 2D scenarios. From the simulations an intriguing observation is made: the solutions generated through the modified tanh method exhibit a singular nature, with some of hybrid waves among them. On contrary to this, solutions derived through the SSEM, tend to be mostly non-singular in nature. The varying influence of the noise intensity revealed that the high amplitude and high energy regions of the waves are more vulnerable to the induced noise as compared to lower energy regions, which are relatively robust. This study introduces novel approaches by incorporating Brownian motion into the extended Kairat-II equation, providing new insights into the behavior of stochastic integrable systems that have not been previously explored.

Linear–quadratic mean-field game for stochastic systems with partial observation

This paper is concerned with a class of linear–quadratic stochastic large-population problems with partial information, where the individual agent only has access to a noisy observation process related to the state. The dynamics of each agent follows a linear stochastic differential equation driven by the individual noise, and all agents are coupled together via the control average term. By studying the associated mean-field game and using the backward separation principle with a state decomposition technique, the decentralized optimal control can be obtained in the open-loop form through a forward–backward stochastic differential equation with the conditional expectation. The optimal filtering equation is also provided. Thanks to the decoupling method, the decentralized optimal control can also be further presented as the feedback of state filtering via the Riccati equation. The explicit solution of the control average limit is given, and the consistency condition system is discussed. Moreover, the related ɛ-Nash equilibrium property is verified. To illustrate the good performance of theoretical results, an example in finance is studied.

On the Approximation of Operator-Valued Riccati Equations in Hilbert Spaces

On the Approximation of Operator-Valued Riccati Equations in Hilbert Spaces

Novel solution behaviors for some communication equation and plasma physics equation via generalized Riccati equation mapping method

Novel solution behaviors for some communication equation and plasma physics equation via generalized Riccati equation mapping method

Learning-based adaptive optimal control of linear time-delay systems: A value iteration approach

This paper proposes a novel learning-based adaptive optimal controller design method for a class of continuous-time linear time-delay systems. A key strategy is to exploit the state-of-the-art reinforcement learning (RL) techniques and adaptive dynamic programming (ADP), and propose a data-driven method to learn the near-optimal controller without the precise knowledge of system dynamics. Specifically, a value iteration (VI) algorithm is proposed to solve the infinite-dimensional Riccati equation for the linear quadratic optimal control problem of time-delay systems using finite samples of input-state trajectory data. It is rigorously proved that the proposed VI algorithm converges to the near-optimal solution. Compared with the previous literature, the nice features of the proposed VI algorithm are that it is directly developed for continuous-time systems without discretization and an initial admissible controller is not required for implementing the algorithm. The efficacy of the proposed methodology is demonstrated by two practical examples of metal cutting and autonomous driving.

Remark on Falaco Soliton as a Tunneling Mechanism in a Navier-Stokes Universe

This paper is a follow up to our previous article [1] suggesting that it is possible to find tunneling time solutions for Schrodinger equation considering quasicrystalline as interstellar matter, by virtue of quasicrystalline potential. The paper also discusses the mapping of these equations to Riccati equations, a class of nonlinear differential equations. This mapping can provide insights into the behavior of the Navier-Stokes equations and may lead to new methods for solving them. The Navier-Stokes equations, a set of nonlinear partial differential equations, are fundamental in fluid mechanics. They describe the motion of viscous fluids. In three dimensions, these equations are particularly complex and often leading to turbulence. The paper also discusses shortly on Falaco soliton as a tunneling mechanism in a Navier-Stokes Universe, which is quite able to fill the gap of realistic mechanism of quantum tunneling which is missing in standard Wave Mechanics. Further investigations are advised.

A fuzzy state-dependent Riccati equation control: adaptive tuning of the state weighting matrix

A fuzzy state-dependent Riccati equation control: adaptive tuning of the state weighting matrix

A Nonlinear Optimal Control Approach for Dual-Arm Robotic Manipulators

Dual-arm robotic manipulators are used in industry and for assisting humans since they enable dexterous handling of objects and more agile and secure execution of pick-and-place, grasping or assembling tasks. In this paper, a nonlinear optimal control approach is proposed for the dynamic model of a dual robotic arm. In the considered application, the dual-arm robotic system has to transfer an object under synchronized motion of its two end-effectors so as to achieve precise positioning and to compensate for contact forces. The dynamic model of this robotic system is formulated while it is proven that the state-space description of the robot’s dynamics is differentially flat. Next, to solve the associated nonlinear optimal control problem, the dynamic model of the dual-arm robot undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on Taylor series expansion and on the associated Jacobian matrices. For the linearized state-space model of the dual-arm robot, a stabilizing optimal (H-infinity) feedback controller is designed. This controller stands for the solution to the nonlinear optimal control problem under model uncertainty and external perturbations. To compute the controller’s feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis. The proposed nonlinear optimal control approach achieves fast and accurate tracking of reference setpoints under moderate variations of the control inputs.

Retrieval of optical soliton patterns for time-fractional nonlinear Schrödinger equation integrating Kudryashov’s law of refractive index with dual form of nonlocal nonlinearity

The Schrödinger model, which describes the mobility of a single wave in an optical fiber, is crucial to communication systems. By using the quadrupled-power law and the dual form of nonlocal nonlinearity, this study concentrates on establishing optical solitary solutions for Kudryashov’s law of nonlinear refractive index. The unified Riccati equation method and the new extended auxiliary equation approach are used to derive numerous ranges of Jacobi elliptic, trigonometric, hyperbolic, and dual-wave solitary solution patterns. These findings will eventually lead to the discovery of bright, singular, kink, dark–bright, and singular-periodic soliton solutions, whose 3D, 2D, and density depictions are provided with the help of mathematical software, Mathematica. Moreover, we provide 2D graphical representations for different values of the fractional and time parameters to show how these novel optical solutions behave. Different features of these solitons arise in many physical contexts such as nonlinear optics, fluid dynamics, and laser physics. The derived results show that the proposed framework presents a rich and varied spectrum of wave phenomena and that the methods for solving nonlinear partial differential equations arising in mathematical physics are effective, useful, and dependable for further exploration in the vast field of optical and mathematical physics.

Unveiling New Exact Solutions of the Complex-Coupled Kuralay System Using the Generalized Riccati Equation Mapping Method

This examination analyzes the integrable dynamics of induced curves by utilizing the complex-coupled Kuralay system (CCKS). The significance of the coupled complex Kuralay equation lies in its role as an essential model that contributes to the understanding of intricate physical and mathematical concepts, making it a valuable tool in scientific research and applications. The soliton solutions originating from the Kuralay equations are believed to encapsulate cutting-edge research in various essential domains such as optical fibers, nonlinear optics, and ferromagnetic materials. Analytical procedures are operated to derive traveling wave solutions for this model, given that the Cauchy problem cannot be resolved using the inverse scattering transform. This study uses the generalized Riccati equation mapping (GREM) method to search for analytical solutions. This method observes single and combined wave solutions in the shock, complex solitary shock, shock singular, and periodic singular forms. Rational solutions also emerged during the derivation. In addition to the analytical results, numerical simulations of the solutions are presented to enhance comprehension of the dynamic features of the solutions generated. The study's conclusions could provide insightful information about how to solve other nonlinear partial differential equations (NLPDEs). The soliton solutions found in this work provide valuable information on the complex nonlinear problem under investigation. These results provide a foundation for further investigation, making the solutions helpful, manageable, and trustworthy for the future development of intricate nonlinear issues. This study's methodology is reliable, robust, effective, and applicable to various NLPDEs. The Maple software application is used to verify the correctness of all obtained solutions.

Exact solutions and bifurcations of the new (3+1)-dimensional generalized Kadomstev-Petviashvili equation

Abstract The exact solutions of the new generalized (3+1)-dimensional Kadomtsev-Petviashvili (ngKP) equation by using the modified extended tanh-function method, the improved ( G ′ / G ) method and the generalized Riccati equation mapping method are considered. Different types of exact traveling wave solutions are obtained, enriching the exact solutions of this equation. In addition, the dynamical behaviors of the different types of solutions obtained by the equation are graphically presented using 3D, 2D and density plots. Then, the phase portraits of the equations are obtained through the bifurcation theory to verify the validity of the previous three methods. Finally, the comparison between our results and the known results is given, it can be concluded that we have provided a large number of new travelling wave solutions to the equation.

ON SENSITIVITY OF SOLUTIONS OF RICCATI EQUATIONS UNDER SMALL PARAMETER PERTURBATIONS AND OPTIMALITY IN LINEAR STOCHASTIC CONTROL SYSTEMS

We investigate sensitivity of solutions of Riccati equations under asymptotically small perturbations of their coefficients. Upper bound on the difference between solutions of algebraic and differential Riccati equations is derived. The result is applied to study optimality in the stochastic linear-quadratic control problem over an infinite time-horizon for an asymptotically autonomous system. We also treat an issue related to performance of invariant control strategy.

Multidimensional Indefinite Stochastic Riccati Equations and Zero-Sum Stochastic Linear-Quadratic Differential Games with Non-Markovian Regime Switching

Multidimensional Indefinite Stochastic Riccati Equations and Zero-Sum Stochastic Linear-Quadratic Differential Games with Non-Markovian Regime Switching

Infinite-time optimal feedback control of a 3-phase asynchronous motor exploiting a nonlinear finite element model

This paper presents modelling of a squirrel-cage induction motor and an optimal control method based on suboptimal control for nonlinear systems to minimise consumed energy and power losses in an induction motor drive. A coupled motor model with optimal control as a closed-loop integrated system is proposed. For modelling of the squirrel-cage asynchronous machine, a field-circuit-mechanical finite-element (FE) model is employed, in which mechanical motion is realised by a moving-mesh method and fixed mesh approach. For the control problem purpose, a surrogate induction motor model, described in a stationary rotor reference d–q frame, is applied. The optimal control is realised by a nonlinear feedback compensator method based on the state-dependent Riccati equation (SDRE) with an infinite time horizon with the surrogate model state-dependent parametrisation (SDP). To perform the control strategy, a SDRE technique with Moore–Penrose pseudoinverse is adopted. To improve the accuracy of the optimisation procedure, a finite element model was used to calculate the motor performance.

Nonlinear Optimal and Multi-Loop Flatness-Basedcontrol of Omnidirectional 3-Wheel Mobile Robots

In this article, the control problem for omnidirectional 3-wheel autonomous mobile robots is solved with the use of (i) a nonlinear optimal control method (ii) a flatness-based control approach which is implemented in successive loops. To apply method (i) that is nonlinear optimal control, the dynamic model of the omnidirectional 3-wheel autonomous mobile robots undergoes approximate linearization at each sampling instant with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrix. The linearization point is defined by the present value of the system's state vector and by the last sampled value of the control inputs vector. To compute the feedback gains of the optimal controller an algebraic Riccati equation is repetitively solved at each time-step of the control algorithm. The global stability properties of the nonlinear optimal control method are proven through Lyapunov analysis. To implement control method (ii), that is flatness-based control in successive loops, the state-space model of the omnidirectional 3-wheel autonomous mobile robot is separated into chained subsystems, which are connected in cascading loops. Each one of these subsystems can be viewed independently as a differentially flat system and control about it can be performed with inversion of its dynamics as in the case of input-output linearized flat systems. The state variables of the preceding (i-th) subsystem become virtual control inputs for the subsequent (i+1-th) subsystem. In turn, exogenous control inputs are applied to the last subsystem. The whole control method is implemented in successive loops and its global stability properties are also proven through Lyapunov stability analysis. The proposed method achieves trajectory tracking and autonomous navigation for the omnidirectional 3-wheel autonomous mobile robots without the need of diffeomorphisms and complicated state-space model transformations.

On Linear Quadratic Optimal Control and Algebraic Riccati Equations for Infinite-Dimensional Differential-Algebraic Equations

We consider linear quadratic optimal control for a very general class of infinite-dimensional differential-algebraic equations (namely, the class of future-resolvable input/state/output nodes) and obtain an algebraic Riccati equation.

Numerical treatment by implementing a successive approximation method for fractional Riccati and logistic differential equations

Numerical treatment by implementing a successive approximation method for fractional Riccati and logistic differential equations

Suboptimal distributed cooperative control for linear multi-agent system via Riccati design.

In this paper, we propose a suboptimal distributed cooperative control scheme for the continuous-time linear multi-agent system (MAS) with a specified global quadratic cost functional over both undirected and directed graph scenarios. For undirected graphs, we first derive the cost functional for a given strictly linear feedback distributed protocol. It is shown that the cost functional is upper bounded by a quadratic form of the MAS's initial state, and the minimum upper bound can be derived by solving a parametric algebraic Riccati equation (PARE) depends solely on the algebraic connectivity of the graph and is independent of the largest eigenvalue compared with the existing work. Based on this, a suboptimal distributed design method is proposed, where the resulting cost functional is less than a specified positive scalar. Then, we extend the theoretical results and design method to the directed graph scenario by introducing the row and column Laplacian matrices associated with the directed graph. Finally, numerical examples are provided to verify the effectiveness of the obtained results.