Showing papers on "Riccati equation published in 2005"
TL;DR: In this article, an exact quantization rule for the Schrodinger equation is presented, in which in addition to Nπ, there is an integral term, called the quantum correction, which is invariant, independent of the number of nodes in the wave function.
Abstract: An exact quantization rule for the Schrodinger equation is presented. In the exact quantization rule, in addition to Nπ, there is an integral term, called the quantum correction. For the exactly solvable systems we find that the quantum correction is an invariant, independent of the number of nodes in the wave function. In those systems, the energy levels of all the bound states can be easily calculated from the exact quantization rule and the solution for the ground state, which can be obtained by solving the Riccati equation. With this new method, we re-calculate the energy levels for the one-dimensional systems with a finite square well, with the Morse potential, with the symmetric and asymmetric Rosen-Morse potentials, and with the first and the second Poschl-Teller potentials, for the harmonic oscillators both in one dimension and in three dimensions, and for the hydrogen atom.
161 citations
TL;DR: In this paper, an extended Fan's sub-equation method is used for constructing exact travelling wave solutions of nonlinear partial differential equations (NLPDEs), where the key idea of this method is to take full advantage of the general elliptic equation involving five parameters, which has more new solutions and whose degeneracies can lead to special subequations involving three parameters.
155 citations
TL;DR: In this paper, the existence of a Lagrangian description for the second-order Riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized Riemannian equation.
Abstract: The existence of a Lagrangian description for the second-order Riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized Riccati equation. The Lagrangians are non-natural and the forces are not derivable from a potential. The constant value E of a preserved energy function can be used as an appropriate parameter for characterizing the behavior of the solutions of these two systems. In the second part the existence of two-dimensional versions endowed with superintegrability is proved. The explicit expressions of the additional integrals are obtained in both cases. Finally it is proved that the orbits of the second system, that represents a nonlinear oscillator, can be considered as nonlinear Lissajous figures
123 citations
TL;DR: In this paper, a semigroup-based definition of the solution of the Gurtin-Pipkin equation with Dirichlet boundary conditions is given, and the dominant term of the input-to-state map is the control to displacement operator of the wave equation.
Abstract: In this paper we give a semigroup-based definition of the solution of the Gurtin-Pipkin equation with Dirichlet boundary conditions. It turns out that the dominant term of the input-to-state map is the control to displacement operator of the wave equation. This operator is surjective if the time interval is long enough. We use this observation in order to prove exact controllability in finite time of the Gurtin-Pipkin equation.
122 citations
TL;DR: In this paper, a new Riccati equation rational expansion method was proposed to uniformly construct a series of exact solutions for nonlinear evolution equations, including rational triangular periodic wave solutions, rational solitary wave solutions and rational wave solutions.
Abstract: In this paper, we present a new Riccati equation rational expansion method to uniformly construct a series of exact solutions for nonlinear evolution equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general solutions. The solutions obtained in this paper include rational triangular periodic wave solutions, rational solitary wave solutions and rational wave solutions. The efficiency of the method can be demonstrated on (2 + 1)-dimensional Burgers equation.
106 citations
TL;DR: This paper is devoted to the study of a stochastic linear-quadratic optimal control problem where the control variable is constrained in a cone, and all the coefficients of the problem are random processes.
Abstract: This paper is devoted to the study of a stochastic linear-quadratic (LQ) optimal control problem where the control variable is constrained in a cone, and all the coefficients of the problem are random processes. Employing Tanaka's formula, optimal control and optimal cost are explicitly obtained via solutions to two extended stochastic Riccati equations (ESREs). The ESREs, introduced for the first time in this paper, are highly nonlinear backward stochastic differential equations (BSDEs), whose solvability is proved based on a truncation function technique and Kobylanski's results. The general results obtained are then applied to a mean-variance portfolio selection problem for a financial market with random appreciation and volatility rates, and with short-selling prohibited. Feasibility of the problem is characterized, and efficient portfolios and efficient frontier are presented in closed forms.
105 citations
TL;DR: The structure-preserving doubling algorithm is developed from a new point of view and its quadratic convergence under assumptions which are weaker than stabilizability and detectability are shown, as well as practical issues involved in the application of the SDA to CAREs.
101 citations
TL;DR: In this article, a new non-linear control synthesis technique (θ-D approximation) is discussed, which achieves suboptimal solutions to a class of nonlinear optimal control problems characterized by a quadratic cost function.
Abstract: In this paper, a new non-linear control synthesis technique (θ–D approximation) is discussed. This approach achieves suboptimal solutions to a class of non-linear optimal control problems characterized by a quadratic cost function and a plant model that is affine in control. An approximate solution to the Hamilton–Jacobi–Bellman (HJB) equation is sought by adding perturbations to the cost function. By manipulating the perturbation terms both semi-global asymptotic stability and suboptimality properties are obtained. The new technique overcomes the large-control-for-large-initial-states problem that occurs in some other Taylor series expansion based methods. Also this method does not require excessive online computations like the recently popular state dependent Riccati equation (SDRE) technique. Furthermore, it provides a closed-form non-linear feedback controller if finite number of terms are taken in the series expansion. A scalar problem and a 2-D benchmark problem are investigated to demonstrate the effectiveness of this new technique. Both stability and convergence proofs are given. Copyright © 2004 John Wiley & Sons, Ltd.
96 citations
TL;DR: In this paper, the authors established sufficient conditions which guarantee that every solution of the third-order nonlinear dynamic equation (c(t)(a(t)xΛ(t))Λ)Λ + q (t)f(x(t))) = 0, t ≥ t0, oscillates or converges to zero.
88 citations
TL;DR: In this article, the extended tanh function method was further improved by generalizing the Riccati equation and picking up its new solutions, and the abundant new non-travelling wave solutions were obtained.
Abstract: The tanh method is used to find travelling wave solutions to various wave equations. In this paper, the extended tanh function method is further improved by generalizing the Riccati equation and picking up its new solutions. In order to test the validity of this approach, the (3 + 1)-dimensional Kadomtsev–Petviashvili equation is considered. As a result, the abundant new non-travelling wave solutions are obtained.
82 citations
TL;DR: Time domain variational analysis is used in a reduction to an open loop differential game, leading to a complete, necessary and sufficient characterization of suboptimal values and an explicit state space design in terms of a parameterized (nonstandard) algebraic matrix Riccati equation in a general continuous time linear system setting.
Abstract: Preview control and fixed-lag smoothing allow a noncausal component in the controller/estimator. Time domain variational analysis is used in a reduction to an open loop differential game, leading to a complete, necessary and sufficient characterization of suboptimal values and an explicit state space design, in terms of a parameterized (nonstandard) algebraic matrix Riccati equation in a general continuous time linear system setting. The solution offers insight into the appropriate structure of the associated Hamiltonian, where the state and co-state are not the usual state of the original dynamic system and that of its adjoint. Rather, the state and co-state are selected to capture the respective lumped effects of initial data and future input selection in the allied game.
TL;DR: An extended Fick-Jacobs equation for the diffusion of noninteracting particles in a two- and symmetric three-dimensional channels of varying cross section A(x) is derived using a variational approach.
Abstract: We derive an extended Fick-Jacobs equation for the diffusion of noninteracting particles in a two- and symmetric three-dimensional channels of varying cross section A(x), using a variational approach. The result is a diffusion differential equation of second order in only one space (longitudinal) coordinate. This equation is tested on the task of calculating the stationary flux through a hyperboloidal tube, and its solution is compared with that of other methods.
TL;DR: This work develops a numerically efficient algorithm for computing controls for nonlinear systems that minimize a quadratic performance measure and shows that each iteration produces a descent direction for the performance measure.
Abstract: We develop a numerically efficient algorithm for computing controls for nonlinear systems that minimize a quadratic performance measure. We formulate the optimal control problem in discrete-time, but many continuous-time problems can be also solved after discretization. Our approach is similar to sequential quadratic programming for finite-dimensional optimization problems in that we solve the nonlinear optimal control problem using sequence of linear quadratic subproblems. Each subproblem is solved efficiently using the Riccati difference equation. We show that each iteration produces a descent direction for the performance measure, and that the sequence of controls converges to a solution that satisfies the well-known necessary conditions for the optimal control.
TL;DR: In this paper, the authors generalized the wellknown non-existence results on two-dimensional Gaussian curvature equation to all-dimensional Q-curvature equation and introduced a new approach to Q-curbature equation which is higher order and even pseudo-differential equation.
TL;DR: This paper considers hybrid controls for a class of linear quadratic problems with white noise perturbation and Markov regime switching, where the regime switching is modeled by a continuous-time Markov chain with a large state space and the control weights are indefinite.
TL;DR: In this paper, a nonlinear control system for the flutter control of an aeroelastic system with unsteady aerodynamics is designed, where both plunge and pitch structural nonlinearities are included.
Abstract: A nonlinear control system for the flutter control of an aeroelastic system with unsteady aerodynamics is designed. The model describes the plunge and pitch motion of a wing. In this model both plunge and pitch structural nonlinearities are included. A single control surface is utilized for the flutter control. For the purpose of design, it is assumed that there exists a specified hard magnitude constraint on the control input. For the synthesis of the controller, only the plunge displacement, pitch angle, and control-surface deflection are measured. The control system design is based on the state-dependent Riccati equation method. A slack variable is introduced to transform the constrained control problem into an unconstrained problem and then a suboptimal nonlinear control law is designed. An observer is constructed to estimate the unavailable state variables of the system for the synthesis of the control system. In the closed-loop system, including the observer and nonlinear controller, the zero state is (locally) asymptotically stable, and the state vector asymptotically converges to the origin. Simulation results for various flow velocities and elastic axis locations are presented, which show that the designed control system is effective in flutter suppression.
TL;DR: It is shown that this computation can be done via computing only the minimal positive solution of a vector equation, which is derived from the special form of solutions of the Riccati equation.
Abstract: We are interested in computing the minimal positive solution of a nonsymmetric algebraic Riccati equation arising in transport theory. We show that this computation can be done via computing only the minimal positive solution of a vector equation, which is derived from the special form of solutions of the Riccati equation. A simple iterative method is presented for solving the vector equation. The simple iteration is much more efficient than the Gauss--Jacobi method presented by Juang in [Linear Algebra Appl., 230 (1995), pp. 89--100] for the Riccati equation. The symmetric case and bounds of the minimal positive solution are also considered. Numerical experiments are given.
TL;DR: Optimal control and optimal value of the model are explicitly obtained based on the solution to a new Riccati-type equation involving both FBM and normal Brownian motion.
Abstract: This paper is concerned with optimal control of stochastic linear systems involving fractional Brownian motion (FBM). First, as a prerequisite for studying the underlying control problems, some new results on stochastic integrals and stochastic differential equations associated with FBM are established. Then, three control models are formulated and studied. In the first two models, the state is scalar-valued and the control is taken as Markovian. Either the problems are completely solved based on a Riccati equation (for model 1, where the cost is a quadratic functional on state and control variables) or optimality is characterized (for model 2, where the cost is a power functional). The last control model under investigation is a general one, where the system involves the Stratonovich integral with respect to FBM, the state is multidimensional, and the admissible controls are not limited to being Markovian. A new Riccati-type equation, which is a backward stochastic differential equation involving both FBM and normal Brownian motion, is introduced. Optimal control and optimal value of the model are explicitly obtained based on the solution to this Riccati-type equation.
01 Jan 2005
TL;DR: In this paper, the authors introduce the concept of stability radius for time-varying linear systems and explore the relationship between the stability radius, the norm of a perturbation operator, and the solvability of a nonstandard differential Riccati equation.
Abstract: This paper introduces the concept of stability radius for time-varying linear systems. Invariance properties of the stability radius are analysed for the group of Bohl transformations. We also explore the relationship between the stability radius, the norm of a certain perturbation operator, and the solvability of a nonstandard differential Riccati equation. As an application we construct robust Lyapunov functions and show how they can be used to analyze robustness with respect to nonlinear perturbations.
TL;DR: In this paper, a new global approach for rendezvous linear quadratic controller design is presented, where instead of solving the algebraic Riccati equation, a continuous simulated annealing (SA) algorithm is used to design rendezvous LQ controller.
TL;DR: These results are proved under a controllability assumption and the proofs rely on general results about the algebraic Riccati equation associated with the linear quadratic regulator problem.
Abstract: We study the wellposedness and the main features of a class of feedback control systems. The involved control system is composed of the generator of a strongly continuous group for the free part and of an unbounded control operator, so that the results can be applied to boundary or point control problems for partial differential equations of hyperbolic or Petrowski type. The feedback operator is explicit and one can achieve an arbitrary large decay rate for the closed-loop system. These results are proved under a controllability assumption and the proofs rely on general results about the algebraic Riccati equation associated with the linear quadratic regulator problem.
Journal Article•
TL;DR: In this paper, the authors used the solutions of forward-backward stochastic differential equations to get the explicit form of the optimal control for linear quadratic stochastically optimal control problem and the open-loop Nash equilibrium point for nonzero sum differential games problem.
Abstract: In this paper, we use the solutions of forward-backward stochastic differential equations to get the explicit form of the optimal control for linear quadratic stochastic optimal control problem and the open-loop Nash equilibrium point for nonzero sum differential games problem. We also discuss the solvability of the generalized Riccati equation system and give the linear feedback regulator for the optimal control problem using the solution of this kind of Riccati equation system.
TL;DR: Two different notions of solutions to BSREs are proposed and it is shown that such solutions allow the synthesis of the optimal control and are proved to have existence and uniqueness results.
Abstract: We study backward stochastic Riccati equations (BSREs) arising in quadratic optimal control problems with infinite dimensional stochastic differential state equations. We allow the coefficients, both in the state equation and in the cost, to be random. In such a context BSREs are backward stochastic differential equations existing in a non-Hilbert space and involving quadratic nonlinearities. We propose two different notions of solutions to BSREs and prove, for both of them, existence and uniqueness results. We also show that such solutions allow us to perform the synthesis of the optimal control. Finally we apply our results to the optimal control of a delay equation and of a wave equation with random damping.
TL;DR: In this article, a variable-coefficient projective Riccati equation method is proposed to solve the generalized Broer-Kaup (SGBK) system, based on a new intermediate transformation.
Abstract: In this paper, based on a new intermediate transformation, a variable-coefficient projective Riccati equation method is proposed. Being concise and straightforward, it is applied to a new (2 + 1)-dimensional simplified generalized Broer–Kaup (SGBK) system. As a result, several new families of exact soliton-like solutions are obtained, beyond the travelling wave. When imposing some condition on them, the new exact solitary wave solutions of the (2 + 1)-dimensional SGBK system are given. The method can be applied to other nonlinear evolution equations in mathematical physics.
TL;DR: In this paper, a robust filter is designed using two sets of coupled Riccati-like equations such that the estimation error is guaranteed to have an upper bound, where the unknown nonlinearities in the system are time varying and norm bounded.
Abstract: This paper considers the problems of stability and filtering for a class of linear hybrid systems with nonlinear uncertainties and Markovian jump parameters. The hybrid system under study involves a continuous-valued system state vector and a discretevalued system mode. The unknown nonlinearities in the system are time varying and norm bounded. The Markovian jump parameters are modeled by a Markov process with a finite number of states. First, we show the equivalence of the sets of norm-bounded linear and nonlinear uncertainties. Then, instead of the original hybrid linear system with nonlinear uncertainties, we consider the same system with linear uncertainties. By using a Riccati equation approach for this new system, a robust filter is designed using two sets of coupled Riccati-like equations such that the estimation error is guaranteed to have an upper bound.
TL;DR: A generalization of the finite-horizon linear quadratic regulator problem is proposed for LTI continuous-time controllable systems, and is based on an algebraic Riccati equation and a Lyapunov equation that enable all the solutions of the Hamiltonian differential equation to be parametrized in closed form.
TL;DR: The Newton method is developed for the vector equation and is more simple and efficient than the corresponding Newton method directly for original Riccati equation and can preserve the form that any solution of the RicCati equation must satisfy.
Abstract: The computation of the minimal positive solution of a non-symmetric algebraic Riccati equation arising in transport theory is considered. It was shown in (SIAM J Matrix Anal Appl, submitted) that this can be done via only computing the minimal positive solution of a vector equation, which is derived from special form of the solutions of the Riccati equation and by exploitation of the special structure of the coefficient matrices of the Riccati equation. In this paper, the Newton method is developed for the vector equation. The Newton method is more simple and efficient than the corresponding Newton method directly for original Riccati equation and can preserve the form that any solution of the Riccati equation must satisfy. Combination of the simple iteration and the Newton iteration is also considered. Numerical examples are given. Copyright © 2004 John Wiley & Sons, Ltd.
TL;DR: The nonlinear decentralized state feedback fuzzy controllers are proposed to stabilize the whole perturbed fuzzy time-delay interconnected system asymptotically and do not need the solution of a Lyapunov equation or a Riccati equation.
08 Jun 2005
TL;DR: In this paper, a nonlinear observer based on a Kalman filter that uses the state dependent Riccati equation (SDRE) to obtain the filter gain is proposed, which does not involve the evaluation of a Jacobian at every time step.
Abstract: A nonlinear asymptotic observer for a discrete-time nonlinear system is considered. The observer is based on a Kalman filter that uses the state dependent Riccati equation (SDRE) to obtain the filter gain. Unlike the extended Kalman filter, the SDRE-based Kalman filter does not involve the evaluation of a Jacobian at every time step. The convergence properties of the SDRE-based Kalman filter when used as an observer in a deterministic setting are analyzed. A few simulation examples are provided to demonstrate the performance and implementation of the SDRE-based observer in both deterministic and stochastic settings.
TL;DR: In this article, the Riccati equation was used to obtain exact solutions for the higher-order nonlinear Schordinger equation in nonlinear optical fibres in a unified way.
Abstract: First, by using the generally projective Riccati equation method, many kinds of exact solutions for the higher-order nonlinear Schordinger equation in nonlinear optical fibres are obtained in a unified way. Then, some relations among these solutions are revealed.