Saddle-Point Equilibrium Strategy for Continuous Singular Bilinear System

In this paper we develop dissipativity results for continual nonlinear and linear singular systems. To the best knowledge of author results are nonexistent. We generalize dissipativity theory to nonlinear continuous singular dynamical systems. Specifically, the classical concepts of system storage functions and supply rates are extended to singular dynamical systems providing a generalized system energy interpretation in terms of stored energy and dissipated energy over the continuous-time system dynamics. Furthermore, extended Kalman-Yakubovich-Popov conditions in terms of the singular system dynamics characterizing dissipativeness via system storage functions are derived. Finally, the framework is specialized to passive and nonexpansive singular systems to provide a generalization of the classical notions of passivity and nonexpansivity for nonlinear singular systems.

This paper is devoted to the controller design based on a fault-tolerant observer for a class of nonlinear continuous singular control systems, consider the sensor failure and time-varying network-induced delay, a new controller and observer co-designing technique are developed, a sufficient condition for the existence of the investiged controller and fault-tolerant observer is derived, and a systematic design approach is addressed by using Lyapunov-Krasovskii function and linear matrix inequalities (LMIs) for controller and observer co-designing. Finally, a numerical example is provided to show the validity of the proposed method.

This paper discusses strict linear matrix inequalities(LMIs) sufficient condition which can be removed of the equality constraint for linear continuous singular systems. The criteria of stability and stabilization of continuous singular system are presented. The conditions are expressed in terms of strict positive definite LMIs which can reduce the equality constraint and semi-positive definite LMIs and are much more tractable and reliable in numerical computation than existing conditions. The conditions make use of only the system parameter matrices (E, A, B) directly by a restricted equivalent transform without introducing other adjective parameters. A counterexample is given to illustrate the criteria are only a sufficient condition but not are a necessary condition. The idea presented can help to find necessary and sufficient strict LMI criteria without equality constraint conditions of stability and stabilization of continuous singular system.

The work develops transformation from regular pencil to standard pencil for a class of continuous singular systems with state and output time delays. It proposes a state feedback control to remove the impulsive modes from the system response. Finally, it presents a new technique for decomposing the time-delay singular system to an equivalent low-order regular system with a direct transmission term from input to output, with a view to developing a high-performance trajectory tracker for the sampled-data time-delay singular system.

Admissibility and robust controller design are considered here for uncertain 2-D continuous singular systems with interval time-varying delays and norm-bounded parameter uncertainties. A new delay-dependent admissibility condition is obtained by using an augmented Lyapunov–Krasovskii's functional and then using the Wirtinger inequality combined with an improved reciprocally convex approach. Then, a robust controller is designed in terms of a linear matrix inequality. Numerical examples are proposed to demonstrate the efficiency of the proposed methods.

We consider a singular control problem that aims to maximize the expected cumulative rewards, where the instantaneous returns depend on the state of a controlled process. The contributions of this paper are twofold. Firstly, to establish sufficient conditions for determining the optimality of the one-barrier strategy when the uncontrolled process X follows a spectrally negative Lévy process with a Lévy measure defined by a completely monotone density. Secondly, to verify the optimality of the (2n+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2n+1)$$\\end{document}-barrier strategy when X is a Brownian motion with a drift. Additionally, we provide an algorithm to compute the barrier values in the latter case.

Infinite horizon singular optimal control problems with control taking values in a closed cone U in Rn lead to a dynamic programming equation of the form: maxF2x,v, v',v'' ,F1x,v,v' =0, forall x∈Q, where Q , the state space of the control problem, is some nonempty connected open subset of Rn , and F1, F2 are continuous real-valued functions on QxR2 and QxR3 respectively, with the coercivity assumption for F 2, that is, the function r&rarrr;F2x,u,p,r is nonincreasing on R . A major concern is to determine how smooth the value function v is across the free boundary of the problem. Linear-convex deterministic and stochastic singular control problems in dimension one are considered. We present the analysis of the above equation together with smoothness of the value function across the free boundary. The interest here is the explicitness of the results and the fact that the smooth fit principle depends on the parameters of the control problem. In particular, the value function for the stochastic control problem is founded explicitly, and at the same time, optimal controls are identified using a verification theorem.

The internal properness and stability of singular decentralized control systems are discussed, and a sufficient and necessary condition that singular systems can be properly stabilized by normal decentralized compensators is presented, i.e. there do not exist any unstable finite fixed modes or impulsive fixed modes in the singular system.

This work generalizes the connection between the optimal singular stochastic control problem and the optimal stopping problem for regime-switching processes. Via the optimal singular stochastic control, the optimal stopping time and the continuation region are characterized. Moreover, we prove the existence of optimal singular stochastic control for a finite horizon singular control problem with the cost function containing the terminal cost. We prove it directly by the compactification method, which is based on an elaborate application of the properties of probability measures over the càdlàg space. Such a problem was left open in Haussmann and Suo [SIAM J. Control Optim., 33 (1995), pp. 916–936] and [SIAM J. Control Optim., 33 (1995), pp. 937–959]). In addition, our compactification method can remove the convexity condition on the coefficients used in Dufour and Miller [SIAM J. Control Optim., 43 (2004), pp. 708–730].

In this paper, we study a bilinear optimal control problem associated to a chemo-repulsion model with linear production term in a bidimensional domain. The existence, uniqueness and regularity of strong solutions of this model are deduced, proving the existence of a global optimal solution. Afterwards, we derive first-order optimality conditions by using a Lagrange multipliers theorem.

An infinite-dimensional bilinear optimal control problem with infinite-time horizon is considered. The associated value function can be expanded in a Taylor series around the equilibrium, the Taylor series involving multilinear forms which are uniquely characterized by generalized Lyapunov equations. A numerical method for solving these equations is proposed. It is based on a generalization of the balanced truncation model reduction method and some techniques of tensor calculus, in order to attenuate the curse of dimensionality. Polynomial feedback laws are derived from the Taylor expansion and are numerically investigated for a control problem of the Fokker-Planck equation. Their efficiency is demonstrated for initial values which are sufficiently close to the equilibrium.

ABSTRACTWe consider a class of singular control problems driven by a double exponential jump diffusion process, which come from the reversible investment problem. In some interesting cases (e.g., the running cost function is given by the so-called Cobb-Douglas production function), we give the explicit solutions to the singular control problem by using the connection between singular control and optimal switching. We solve a collection of consistent optimal switching problems and yield the explicit solution for the singular control problem. We then give an application to a particular inventory control problem in a single random period.

This report summarizes some of our recent work (Guo and Tomecek, SIAM J Control Optim 47(1):421–443, 2008; A class of singular control problems and smooth fit principle, 2008) on a new theoretical connection between singular control of finite variation and optimal switching problems. This correspondence not only provides a novel method for analyzing multi-dimensional singular control problems, but also builds links among singular controls, Dynkin games, and sequential optimal stopping problems.

A linear quadratic regulator problem where a small parameter μ2multiplies the control cost is considered. As \mu \rightarrow 0 , it exhibits a dynamic behavior with multiple time-scales, whereas for \mu = 0 , it is a singular control problem. At first a multiparameter singularly perturbed model of the given problem is constructed by transferring the singularity that exists in the performance index to the dynamic equations. Then the problem is decomposed into several subproblems of minimal order, each pertaining to only one time-scale. The method developed permits under a single framework the characterization of asymptotic behavior of optimal closed-loop poles, state and control trajectories, performance index, and optimal transfer function as \mu \rightarrow 0 . Moreover, the asymptotic solution brings into focus 1) the nature of singular control and its connection to transmission zeros and 2) the presence of various types of impulses and other higher order distributions in both state and control. The approach unifies and extends significantly several ideas previously somewhat disjoint.

A N IMPORTANT problem in air traffic management (ATM) is the design of aircraft trajectories that meet certain arrival time constraints at given waypoints, for instance, at the top of descent (TOD), at the initial approach fix (IAF), or at the runway threshold. These are called 4-D trajectories, which are a key element in the trajectory-based-operations (TBO) concept proposed by NextGen and SESAR for the future ATM system. The time constraints must also be met in certain cases in which the nominal trajectories have to be modified to resolve detected conflicts (e.g., lost of separation minima) between aircraft; for example, Bilimoria and Lee [1] analyze aircraft conflict resolution with an arrival time constraint at a downstream waypoint. On the other hand, the design of fuel-optimal trajectories that lead to energy-efficient flights is another important problem which has been treated extensively in the literature, see, for instance, Burrows [2], Neuman andKreindler [3],Menon [4] and the references therein. Fuel-optimal trajectories with fixed arrival times are studied by Sorensen and Waters [5], Burrows [6] and Chakravarty [7], who analyze the 4-D fuel-optimization problem as a minimum directoperating-cost (DOC) problem with free final time, that is, the problem is to find the time cost for which the corresponding free final-time DOC-optimal trajectory arrives at the assigned time. In this work we analyze the problem of minimum-fuel cruise at constant altitudewith afixed arrival time as a singular optimal control problem, building upon the works of Pargett and Ardema [8] and Rivas and Valenzuela [9], who analyze the problem of maximumrange cruise at constant altitude also as a singular optimal control problem; the case of unsteady cruise is considered. The singular arcs and the corresponding optimal control are obtained as a function of the final time. The optimal paths are obtained as well, which define a variable-Mach cruise at constant altitude. The influence of cruise altitude on the optimal paths is analyzed, and the minimum fuel is calculated. The final-time constraint may be defined, for example, by a flight delay imposed on the nominal (preferred) cruise trajectory (which in our case is the minimum-fuel cruise trajectory with free final time); comparison with a standard constant-Mach procedure to absorb delays is made. Results are presented for a model of a Boeing 767-300ER. Problem Formulation