Optimal Control for Switched Systems with Pre-defined Order and Switch-Dependent Dynamics

The optimal control problem for switched systems, with a predefined order of switches, is considered. The differential equations may depend explicitly on previous switching instants, and the latter may be state dependent. The solution is based on the Calculus-of-Variations, which leads to a single two-point boundary-value problem. A new condition for the Hamiltonian jump at the switching instants is obtained. Simple numerical examples demonstrate the results.     

Energy dissipating hybrid control for impulsive dynamical systems

A novel class of fixed-order, energy-based hybrid controllers is proposed as a means for achieving enhanced energy dissipation in nonsmooth Euler–Lagrange, hybrid port-controlled Hamiltonian, and lossless impulsive dynamical systems. These dynamic controllers combine a logical switching architecture with hybrid dynamics to guarantee that the system plant energy is strictly decreasing across switchings. The general framework leads to hybrid closed-loop systems described by impulsive differential equations. Special cases of energy-based hybrid controllers involving state-dependent switching are described, and an illustrative numerical example is given to demonstrate the efficacy of the proposed approach.     

大型不正定陀螺系统本征值问题

陀螺动力系统可以导入哈密顿辛几何体系,在哈密顿陀螺系统的辛子空间迭代法的基础上提出了一种能够有效计算大型不正定哈密顿函数的陀螺系统本征值问题的算法．利用陀螺矩阵既为哈密顿矩阵而本征值又是纯虚数或零的特点,将对应哈密顿函数为负的本征值分离开来,构造出对应哈密顿函数全为正的本征值问题,利用陀螺系统的辛子空间迭代法计算出正定哈密顿矩阵的本征值,从而解决了大型不正定陀螺系统的本征值问题,算例证明,本征解收敛得很快．     

Near-optimal control of a stochastic vegetation-water system with reaction diffusion

In this article, we study the near-optimal control of a class of stochastic vegetation-water model. The near-optimal control is one problem in which the density of vegetation and water is higher at the lowest cost. We have provided a priori estimates of the vegetation and water densities and obtained the sufficient and necessary conditions for the system's near-optimal control problem by applying the maximum condition of the Hamiltonian function and the Ekeland principle. A numerical simulation is presented to verify our theoretical results.     

Switching design for suboptimal guaranteed cost control of 2-D nonlinear switched systems in the Roesser model

This paper deals with the problem of switching design for guaranteed cost control of discrete-time two-dimensional (2-D) nonlinear switched systems described by the Roesser model. The switching signal, which determines the active mode of the system, is subject to a state-dependent process whose values belong to a finite index set. By using 2-D common Lyapunov function approach, a sufficient condition expressed in terms of tractable matrix inequalities is first established to design a min-projection switching rule that makes the 2-D switched system asymptotically stable. The obtained result on stability analysis is then utilized to synthesize a suboptimal state feedback controller that minimizes the upper bound of a given infinite-horizon cost function. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design method.     

Finite-time stability and stabilization of nonlinear stochastic hybrid systems

This paper deals with the problem of finite-time stability and stabilization of nonlinear Markovian switching stochastic systems which exist impulses at the switching instants. Using multiple Lyapunov function theory, a sufficient condition is established for finite-time stability of the underlying systems. Furthermore, based on the state partition of continuous parts of systems, a feedback controller is designed such that the corresponding impulsive stochastic closed-loop systems are finite-time stochastically stable. A numerical example is presented to illustrate the effectiveness of the proposed method.     

Noisy Hamiltonian Monte Carlo for Doubly Intractable Distributions

Hamiltonian Monte Carlo (HMC) has been progressively incorporated within the statistician’s toolbox as an alternative sampling method in settings when standard Metropolis–Hastings is inefficient. HMC generates a Markov chain on an augmented state space with transitions based on a deterministic differential flow derived from Hamiltonian mechanics. In practice, the evolution of Hamiltonian systems cannot be solved analytically, requiring numerical integration schemes. Under numerical integration, the resulting approximate solution no longer preserves the measure of the target distribution, therefore an accept–reject step is used to correct the bias. For doubly intractable distributions—such as posterior distributions based on Gibbs random fields—HMC suffers from some computational difficulties: computation of gradients in the differential flow and computation of the accept–reject proposals poses difficulty. In this article, we study the behavior of HMC when these quantities are replaced by Monte Carlo estimates. Supplemental codes for implementing methods used in the article are available online.     

Long-time averaging for integrable Hamiltonian dynamics

Summary. Given a Hamiltonian dynamical system, we address the question of computing the limit of the time-average of an observable. For a completely integrable system, it is known that ergodicity can be characterized by a diophantine condition on its frequencies and that this limit coincides with the space-average over an invariant manifold. In this paper, we show that we can improve the rate of convergence upon using a filter function in the time-averages. We then show that this convergence persists when a symplectic numerical scheme is applied to the system, up to the order of the integrator.     

An inverse‐free ADI algorithm for computing Lagrangian invariant subspaces

The numerical computation of Lagrangian invariant subspaces of large‐scale Hamiltonian matrices is discussed in the context of the solution of Lyapunov equations. A new version of the low‐rank alternating direction implicit method is introduced, which, in order to avoid numerical difficulties with solutions that are of very large norm, uses an inverse‐free representation of the subspace and avoids inverses of ill‐conditioned matrices. It is shown that this prevents large growth of the elements of the solution that may destroy a low‐rank approximation of the solution. A partial error analysis is presented, and the behavior of the method is demonstrated via several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Stochastic discrete Hamiltonian variational integrators

Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge–Kutta methods as a special case. Several new low-stage stochastic symplectic methods of mean-square order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to nonsymplectic methods.     

Stabilization for a class of second-order switched systems

This paper considers the asymptotic stabilization problem of second-order linear time-invariant (LTI) autonomous switched systems consisting of two subsystems with unstable focus equilibrium. More precisely, we find a necessary and sufficient condition for the origin to be asymptotically stable under the predesigned switching law. The result is obtained without looking for a common Lyapunov function or multiple Lyapunov function, but studying the locus in which the two subsystem's vector fields are parallel. Then the “most stabilizing” switching laws are designed which have translated the switched system into a piecewise linear system. Two numerical examples are presented to show the potential of the proposed techniques.     

Hamiltonian Interpolation of Splitting Approximations for Nonlinear PDEs

We consider a wide class of semilinear Hamiltonian partial differential equations and their approximation by time splitting methods. We assume that the nonlinearity is polynomial, and that the numerical trajectory remains at least uniformly integrable with respect to an eigenbasis of the linear operator (typically the Fourier basis). We show the existence of a modified interpolated Hamiltonian equation whose exact solution coincides with the discrete flow at each time step over a long time. While for standard splitting or implicit–explicit schemes, this long time depends on a cut-off condition in the high frequencies (CFL condition), we show that it can be made exponentially large with respect to the step size for a class of modified splitting schemes.     

Index theory, nontrivial solutions, and asymptotically linear second-order Hamiltonian systems

In this paper, we consider the existence and multiplicity of solutions of second-order Hamiltonian systems. We propose a generalized asymptotically linear condition on the gradient of Hamiltonian function, classify the linear Hamiltonian systems, prove the monotonicity of the index function, and obtain some new conditions on the existence and multiplicity for generalized asymptotically linear Hamiltonian systems by global analysis methods such as the Leray-Schauder degree theory, the Morse theory, the Ljusternik-Schnirelman theory, etc.     

无穷维Hamilton算子的二次数值域

研究了一类无界无穷维Hamilton算子的二次数值域的性质,进而,应用二次数值域来刻画了无穷维Hamilton算子谱的分布范围,并给出了二次数值域的闭包包含谱集的结论.     

A Switching Algorithm Based on Modified Quasi-Newton Equation

In this paper, a switching method for unconstrained minimization is proposed. The method is based on the modified BFGS method and the modified SR1 method. The eigenvalues and condition numbers of both the modified updates are evaluated and used in the switching rule. When the condition number of the modified SR1 update is superior to the modified BFGS update, the step in the proposed quasi-Newton method is the modified SR1 step. Otherwise the step is the modified BFGS step. The efficiency of the proposed method is tested by numerical experiments on small, medium and large scale optimization. The numerical results are reported and analyzed to show the superiority of the proposed method.     

线性Hamilton系统边值问题的保辛数值方法

以Hamilton系统的正则变换和生成函数为基础研究线性时变Hamilton系统边值问题的保辛数值求解算法.根据第二类生成函数系数矩阵与状态传递矩阵的关系,构造了生成函数系数矩阵的区段合并递推算法,并进一步将递推算法推广到线性非齐次边值问题中;然后利用生成函数的性质将边值问题转化为初值问题,最后采用初值问题的保辛算法求解以达到整个Hamilton系统保辛的目的.数值算例表明该方法能够有效地求解线性齐次与非齐次问题,并能很好地保持Hamilton系统的固有特性.     

Numerical simulation and geometrical analysis on the onset of chaos in a system of two coupled pendulums

By means of two geometrical approaches: (i) the potential landscape, and (ii) the Hamiltonian manifold method, we have found the relations that predict the transition to chaos in a two-coupled pendulum system due to a shift from positive to negative curvature. Our scrutiny from numerical simulations show that the splitting of the KAM island at the symmetric mode is another scenario that hints at the incidence of chaotic behaviour. We have uncovered two regimes on the onset of chaos, depending on whether the coupling parameter is greater or less than a critical point. In the first regime, chaos is observed to first occur after the splitting of the KAM island at the symmetric mode and before the switching of the dynamical curvature. On the other hand, an interchange of scenarios arises in the second regime, with chaos now appears after curvature switching and before KAM island splitting at the symmetric mode.     

Conditions for realization of a function on a semilattice over real bases of switching elements

The functions are considered on finite upper semilattices that realize their switching networks. Some necessary condition is given for realization of a function on a semilattice over a base of switching elements as well as necessary and sufficient conditions for realization of a function on a semilattice of states by single-stage and many-stage circuits over bases of real switching elements (transistors and resistors).     

Euler-Lagrange and Hamiltonian formalisms in dynamic optimization

We consider dynamic optimization problems for systems governed by differential inclusions. The main focus is on the structure of and interrelations between necessary optimality conditions stated in terms of Euler-Lagrange and Hamiltonian formalisms. The principal new results are: an extension of the recently discovered form of the Euler-Weierstrass condition to nonconvex valued differential inclusions, and a new Hamiltonian condition for convex valued inclusions. In both cases additional attention was given to weakening Lipschitz type requirements on the set-valued mapping. The central role of the Euler type condition is emphasized by showing that both the new Hamiltonian condition and the most general form of the Pontriagin maximum principle for equality constrained control systems are consequences of the Euler-Weierstrass condition. An example is given demonstrating that the new Hamiltonian condition is strictly stronger than the previously known one.

     

Canonical B-series

Summary. B-series provide a powerful general tool to express numerical methods for differential equations. Many differential equations are of Hamiltonian form and there has been much recent interest in constructing so-called canonical or symplectic integrators for the Hamiltonian case. In this paper we provide a necessary and sufficient condition for a B-series to correspond to a canonical method. Received March 15, 1993