36,415 research outputs found
Branch-and-lift algorithm for deterministic global optimization in nonlinear optimal control
This paper presents a branch-and-lift algorithm for solving optimal control problems with smooth nonlinear dynamics and potentially nonconvex objective and constraint functionals to guaranteed global optimality. This algorithm features a direct sequential method and builds upon a generic, spatial branch-and-bound algorithm. A new operation, called lifting, is introduced, which refines the control parameterization via a Gram-Schmidt orthogonalization process, while simultaneously eliminating control subregions that are either infeasible or that provably cannot contain any global optima. Conditions are given under which the image of the control parameterization error in the state space contracts exponentially as the parameterization order is increased, thereby making the lifting operation efficient. A computational technique based on ellipsoidal calculus is also developed that satisfies these conditions. The practical applicability of branch-and-lift is illustrated in a numerical example. © 2013 Springer Science+Business Media New York
A Gauss pseudospectral transcription for optimal control
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2005.Includes bibliographical references (p. 237-243).A pseudospectral method for solving nonlinear optimal control problems is proposed in this thesis. The method is a direct transcription that transcribes the continuous optimal control problem into a discrete nonlinear programming problem (NLP), which can be solved by well-developed algorithms. The method is based on using global polynomial approximations to the dynamic equations at a set of Gauss collocation points. The optimality conditions of the NLP have been found to be equivalent to the discretized optimality conditions of the continuous control problem, which is not true of other pseudospectral methods. This result indicates that the method can take advantage of the properties of both direct and indirect formulations, and allows for the costates to be estimated directly from the Lagrange multipliers of the NLP. The method has been shown empirically to have very fast convergence (exponential) in the states, controls, and costates, for problems with analytic solutions. This convergence rate of the proposed method is significantly faster than traditional finite difference methods, and has been demonstrated with many example problems. The initial costate estimate from the proposed method can be used to define an optimal feedback law for real time optimal control of nonlinear problems. The application and effectiveness of this approach has been demonstrated with the simulated trajectory optimization of a launch vehicle.by David Benson.Ph.D
A Weighted Residual Framework for Formulation and Analysis of Direct Transcription Methods for Optimal Control
In the past three decades, numerous methods have been proposed to transcribe optimal control problems (OCP) into nonlinear programming problems (NLP). In this dissertation work, a unifying weighted residual framework is developed under which most of the existing transcription methods can be derived by judiciously choosing test and trial functions. This greatly simplifies the derivation of optimality conditions and costate estimation results for direct transcription methods. Under the same framework, three new transcription methods are devised which are particularly suitable for implementation in an adaptive refinement setting. The method of Hilbert space projection, the least square method for optimal control and generalized moment method for optimal control are developed and their optimality conditions are derived. It is shown that under a set of equivalence conditions, costates can be estimated from the Lagrange multipliers of the associated NLP for all three methods. Numerical implementation of these methods is described using B-Splines and global interpolating polynomials as approximating functions. It is shown that the existing pseudospectral methods for optimal control can be formulated and analyzed under the proposed weighted residual framework. Performance of Legendre, Gauss and Radau pseudospectral methods is compared with the methods proposed in this research. Based on the variational analysis of first-order optimality conditions for the optimal control problem, an posteriori error estimation procedure is developed. Using these error estimates, an h-adaptive scheme is outlined for the implementation of least square method in an adaptive manner. A time-scaling technique is described to handle problems with discontinuous control or multiple phases. Several real-life examples were solved to show the efficacy of the h-adaptive and time-scaling algorithm
Necessary conditions of first-order for an optimal boundary control problem for viscous damage processes in 2D
Controlling the growth of material damage is an important engineering
task with plenty of real world applications. In this paper we approach this
topic from the mathematical point of view by investigating an optimal
boundary control problem for a damage phase-field model for viscoelastic
media. We consider non-homogeneous Neumann data for the displacement field
which describe external boundary forces and act as control variable. The
underlying hyberbolic-parabolic PDE system for the state variables exhibit
highly nonlinear terms which emerge in context with damage processes. The
cost functional is of tracking type, and constraints for the control variable
are prescribed. Based on recent results from [4], where global-in-time
well-posedness of strong solutions to the lower level problem and existence
of optimal controls of the upper level problem have been established, we show
in this contribution differentiability of the control-to-state mapping,
wellposedness of the linearization and existence of solutions of the adjoint
state system. Due to the highly nonlinear nature of the state system which
has by our knowledge not been considered for optimal control problems in the
literature, we present a very weak formulation and estimation techniques of
the associated adjoint system. For mathematical reasons the analysis is
restricted here to the two-dimensional case. We conclude our results with
first-order necessary optimality conditions in terms of a variational
inequality together with PDEs for the state and adjoint state system
Optimal Controls for Forward-Backward Stochastic Differential Equations: Time-Inconsistency and Time-Consistent Solutions
This paper is concerned with an optimal control problem for a
forward-backward stochastic differential equation (FBSDE, for short) with a
recursive cost functional determined by a backward stochastic Volterra integral
equation (BSVIE, for short). It is found that such an optimal control problem
is time-inconsistent in general, even if the cost functional is reduced to a
classical Bolza type one as in Peng [50], Lim-Zhou [41], and Yong [74].
Therefore, instead of finding a global optimal control (which is
time-inconsistent), we will look for a time-consistent and locally optimal
equilibrium strategy, which can be constructed via the solution of an
associated equilibrium Hamilton-Jacobi-Bellman (HJB, for short) equation. A
verification theorem for the local optimality of the equilibrium strategy is
proved by means of the generalized Feynman-Kac formula for BSVIEs and some
stability estimates of the representation for parabolic partial differential
equations (PDEs, for short). Under certain conditions, it is proved that the
equilibrium HJB equation, which is a nonlocal PDE, admits a unique classical
solution. As special cases and applications, the linear-quadratic problems, a
mean-variance model, a social planner problem with heterogeneous Epstein-Zin
utilities, and a Stackelberg game are briefly investigated. It turns out that
our framework can cover not only the optimal control problems for FBSDEs
studied in [50,41,74], and so on, but also the problems of the general
discounting and some nonlinear appearance of conditional expectations for the
terminal state, studied in Yong [75,77] and Bj\"{o}rk-Khapko-Murgoci [7]
Strong Stationarity Conditions for Optimal Control of Hybrid Systems
We present necessary and sufficient optimality conditions for finite time
optimal control problems for a class of hybrid systems described by linear
complementarity models. Although these optimal control problems are difficult
in general due to the presence of complementarity constraints, we provide a set
of structural assumptions ensuring that the tangent cone of the constraints
possesses geometric regularity properties. These imply that the classical
Karush-Kuhn-Tucker conditions of nonlinear programming theory are both
necessary and sufficient for local optimality, which is not the case for
general mathematical programs with complementarity constraints. We also present
sufficient conditions for global optimality.
We proceed to show that the dynamics of every continuous piecewise affine
system can be written as the optimizer of a mathematical program which results
in a linear complementarity model satisfying our structural assumptions. Hence,
our stationarity results apply to a large class of hybrid systems with
piecewise affine dynamics. We present simulation results showing the
substantial benefits possible from using a nonlinear programming approach to
the optimal control problem with complementarity constraints instead of a more
traditional mixed-integer formulation.Comment: 30 pages, 4 figure
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