Riccati scheme for integrating nonlinear systems of differential equations

A modification of the Riccati system that makes it possible to reduce to linear problems the initial-value problem for systems of ordinary differential equations with bilinear nonlinearity is discussed. It is shown that from the algebraic point of view it is natural, in the framework of the scheme, to consider functions that take values in an algebra with two multiplications related by a condition of the type of associativity.State University, St. Petersburg. Translated from Teoretcheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 352–363, March, 1995.     

Impulsive consensus of one-sided Lipschitz nonlinear multi-agent systems with Semi-Markov switching topologies

Many real systems involve not only parameter changes but also sudden variations in environmental conditions, which often causes unpredictable topologies switching. This paper investigates the impulsive consensus problem of the one-sided Lipschitz nonlinear multi-agent systems (MASs) with Semi-Markov switching topologies. Different from the existing modeling methods of the Markov chain, the Semi-Markov chain is adopted to describe this kind of randomly occurring changes reasonably. To cope with the communication and control cost constraints in the multi-agent systems, the distributed impulsive control method is applied to address the leader–follower consensus problem. Beyond that, to obtain a wider nonlinear application range, the one-sided condition is delicately developed to the controller design, and the results are different from the ones obtained in the traditional method with the Lipschitz condition (note that the existing results are usually only applicable to the case with small Lipschitz constant). Based on the characteristics of cumulative distribution functions, the theory of Lyapunov-like function and impulsive differential equation, the asymptotically mean square consensus of multi-agent systems is maintained with the proposed impulsive control protocol. Finally, an explanatory simulation is presented to validate the correctness of the proposed approach conclusively.     

General observers for discrete-time nonlinear systems

This paper is a geometric study of finding general exponential observers for discrete-time nonlinear systems. Using center manifold theory for maps, we derive necessary and sufficient conditions for general exponential observers for Lyapunov stable discrete-time nonlinear systems. As an application of our characterization of general exponential observers, we give a construction procedure for identity exponential observers for discrete-time nonlinear systems.     

Global Lipschitz regularizing effects for linear and nonlinear parabolic equations

In this paper we prove global bounds on the spatial gradient of viscosity solutions to second order linear and nonlinear parabolic equations in (0,T)×R^N$(0,T)\times {\mathbb{R}}^N$. Our assumptions include the case that the coefficients be both unbounded and with very mild local regularity (possibly weaker than the Dini continuity), the estimates only depending on the parabolicity constant and on the modulus of continuity of coefficients (but not on their L^∞$L^{\infty }$-norm). Our proof provides the analytic counterpart to the probabilistic proof used in Priola and Wang (2006) [35] (J. Funct. Anal. 2006) to get this type of gradient estimates in the linear case. We actually extend such estimates to the case of possibly unbounded data and solutions as well as to the case of nonlinear operators including Bellman–Isaacs equations. We investigate both the classical regularizing effect (at time t>0$t>0$) and the possible conservation of Lipschitz regularity from t=0$t=0$, and similarly we prove global Hölder estimates under weaker assumptions on the coefficients. The estimates we prove for unbounded data and solutions seem to be new even in the classical case of linear equations with bounded and Hölder continuous coefficients. Applications to Liouville type theorems are also given in the paper. Finally, we compare in an appendix the analytic and the probabilistic approach discussing the analogy between the doubling variables method of viscosity solutions and the probabilistic coupling method.     

Riccati equations for generalized singular estimate control systems

We consider the Bolza problem associated with boundary/point control systems governed by strongly continuous semigroups. In continuation of our work in Lasiecka and Tuffaha [I. Lasiecka and A. Tuffaha, Riccati equations for the Bolza problem arising in boundary/point control problems governed by C ₀–semigroups satisfying a singular estimate, J. Optim. Theory Appl. 136 (2008), pp. 229–246; I. Lasiecka and A. Tuffaha, A Bolza optimal synthesis problem for singular estimate control systems, Control Cybernet 38(4B) (2009), pp. 1429–1460], we yet extend the theory to a more general class of control problems that are not analytic providing sharp blow-up rates for the regularity. Solvability of the associated Riccati equations and an optimal feedback synthesis are established. The presence of unbounded control actions, such as boundary/point controls, naturally lead to a singularity at the terminal point t?=?T of the optimal control and of the corresponding feedback operator as before. The class of control systems considered in this article is a generalization to the class usually referred to in the literature as ‘Singular Estimate Control Systems’. The prototype is still that of a PDE system consisting of coupled hyperbolic parabolic dynamics interacting on an interface with point/boundary control. The distinct feature of the class considered in this article is that the degree of unboundedness in the control is stronger than that allowed in the usual singular estimate control system configuration, giving rise to less regular optimal state trajectories.     

New results on general observers for nonlinear systems

In this paper, we establish that detectability is a necessary condition for the existence ofgeneral observers (asymptotic or exponential) for nonlinear systems. Using this necessary condition, we show that there does not exist any general observer (asymptotic or exponential), for nonlinear systems with real parametric uncertainty, if the state equilibrium does not change with the parameter values and if the plant output function is purely a function of the state. Next, using center manifold theory, we derive necessary and sufficient conditions for the existence of general exponential observers for Lyapunov stable nonlinear systems. As an application of this result, we show that for the existence of general exponential observers for Lyapunov stable nonlinear systems, the dimension of the state of the general exponential observer should not be less than the number of critical eigenvalues of the linearization matrix of the state dynamics of the plant.     

Strong invariance and one-sided Lipschitz multifunctions

This paper studies the strong invariance property of a differential inclusion in finite dimensions under the assumption of a locally one-sided Lipschitz condition.     

Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations

We establish interior Lipschitz regularity for continuous viscosity solutions of fully nonlinear, conformally invariant, degenerate elliptic equations. As a by-product of our method, we also prove a weak form of the strong comparison principle, which we refer to as the principle of propagation of touching points, for operators of the form \(\nabla ^2 \psi + L(x,\psi ,\nabla \psi )\) which are non-decreasing in \(\psi \).     

New results on general observers for discrete-time nonlinear systems

In this paper, we establish that detectability is a necessary condition for the existence of general observers (asymptotic or exponential) for discrete-time nonlinear systems. Using this necessary condition, we show that there does not exist any general observer (asymptotic or exponential) for discrete-time nonlinear systems with real parametric uncertainty, if the state equilibrium does not change with the parameter values and if the plant output function is purely a function of the state. Next, using center manifold theory for maps, we derive necessary and sufficient conditions for the existence of general exponential observers for Lyapunov stable discrete-time nonlinear systems. As an application of this result, we show that for the existence of general exponential observers for Lyapunov stable discrete-time nonlinear systems, the dimension of the state of the general exponential observer should not be less than the number of critical eigenvalues of the linearization matrix of the state dynamics of the plant.     

Synchronization for chaotic systems and chaos-based secure communications via both reduced-order and step-by-step sliding mode observers

This paper considers the problems of the chaos synchronization and chaos-based secure communication when the observer matching condition is not satisfied. An auxiliary drive signal vector which may satisfy the observer matching condition is constructed. By using the drive signals of original system, a step-by-step sliding mode observer is considered to obtain the exact estimates of the auxiliary drive signals and their derivatives. A reduced-order observer is designed to asymptotically estimate the states of the drive system. By using the estimates of states and the derivatives of the auxiliary signals, an information signal recovery method which does not use any derivative information of original drive system is developed. Finally, a numerical simulation example is given to illustrate the effectiveness of the proposed methods.     

State observers for a class of multi-output nonlinear dynamic systems

This note considers the problem of observer design for a class of multi-output nonlinear systems. A new state observer design methodology for linear time-varying multi-output systems is presented. Furthermore, we show that the same methodology can be extended to a class of multi-output nonlinear systems. Some sufficient conditions for the existence of the proposed observer are obtained, which guarantee that the error of state estimation converges asymptotically to zero. An example is given to demonstrate the effectiveness of the proposed methodology.     

Optimal solution of nonlinear equations satisfying a Lipschitz condition

Summary For a given nonnegative we seek a pointx ^* such that |f(x ^*)| wheref is a nonlinear transformation of the cubeB=[0,1] ^m into (or ^p ,p>1) satisfying a Lipschitz condition with the constantK and having a zero inB.The information operator onf consists ofn values of arbitrary linear functionals which are computed adaptively. The pointx ^* is constructed by means of an algorithm which is a mapping depending on the information operator. We find an optimal algorithm, i.e., algorithm with the smallest error, which usesn function evaluations computed adaptively. We also exhibit nearly optimal information operators, i.e., the linear functionals for which the error of an optimal algorithm that uses them is almost minimal. Nearly optimal information operators consists ofn nonadaptive function evaluations at equispaced pointsx _j in the cubeB. This result exhibits the superiority of the T. Aird and J. Rice procedure ZSRCH (IMSL library [1]) over Sobol's approach [7] for solving nonlinear equations in our class of functions. We also prove that the simple search algorithm which yields a pointx ^*=x _k such that is nearly optimal. The complexity, i.e., the minimal cost of solving our problem is roughly equal to (K/) ^m .     

Asymptotic observers for a class of evolution equations. A nonlinear approach

We consider in a Hilbert space H the system (E_u) = x = uAx+B(x); y = 〈x. c〉_H, where the control u ε L^∞([0, + ∞[, ℝ⁺) multiplies a possibly unbounded m-dissipative linear operator A. The operator B is nonlinear dissipative, and y stands for the output of the system. We prove, in this nonlinear framework, the existence of a suitable Luenberger-like observer. For this purpose, we show that the usual notions of regularly persistent inputs proposed in [7] or [4] are the appropriate concepts that allow one to generalize the main results of [9] and [8] or [7] for bilinear systems to our nonlinear general system: For each regularly persistent input, the estimation error of the observer converges weakly to zero. If in addition A generates a compact semigroup, the estimation error converges strongly to zero. A prototype of such a system is the heat exchanger system described in [9] or [8].     

Some computational methods for systems of nonlinear equations and systems of polynomial equations

This paper gives a brief survey and assessment of computational methods for finding solutions to systems of nonlinear equations and systems of polynomial equations. Starting from methods which converge locally and which find one solution, we progress to methods which are globally convergent and find an a priori determinable number of solutions. We will concentrate on simplicial algorithms and homotopy methods. Enhancements of published methods are included and further developments are discussed.     

Sliding mode high-gain observers for a class of uncertain nonlinear systems

A robust high-gain observer for state and unknown input estimations for a special class of single-output nonlinear systems is developed in this article. Ensuring the observability of the unknown input with respect to the output, the disturbance can be estimated from the sliding surface. In the sliding mode, the convergence of the estimation error dynamics is proven similar to the analysis of high-gain observers.     

Comparison theory for Riccati equations

We give a short new proof for the comparison theory of the matrix valued Riccati equationB′+B ²+R=0 with singular initial values. Applications to Riemannian geometry are briefly indicated.