2,077 research outputs found
The turnpike property in finite-dimensional nonlinear optimal control
Turnpike properties have been established long time ago in finite-dimensional
optimal control problems arising in econometry. They refer to the fact that,
under quite general assumptions, the optimal solutions of a given optimal
control problem settled in large time consist approximately of three pieces,
the first and the last of which being transient short-time arcs, and the middle
piece being a long-time arc staying exponentially close to the optimal
steady-state solution of an associated static optimal control problem. We
provide in this paper a general version of a turnpike theorem, valuable for
nonlinear dynamics without any specific assumption, and for very general
terminal conditions. Not only the optimal trajectory is shown to remain
exponentially close to a steady-state, but also the corresponding adjoint
vector of the Pontryagin maximum principle. The exponential closedness is
quantified with the use of appropriate normal forms of Riccati equations. We
show then how the property on the adjoint vector can be adequately used in
order to initialize successfully a numerical direct method, or a shooting
method. In particular, we provide an appropriate variant of the usual shooting
method in which we initialize the adjoint vector, not at the initial time, but
at the middle of the trajectory
Low-Thrust Lyapunov to Lyapunov and Halo to Halo with -Minimization
In this work, we develop a new method to design energy minimum low-thrust
missions (L2-minimization). In the Circular Restricted Three Body Problem, the
knowledge of invariant manifolds helps us initialize an indirect method solving
a transfer mission between periodic Lyapunov orbits. Indeed, using the PMP, the
optimal control problem is solved using Newton-like algorithms finding the zero
of a shooting function. To compute a Lyapunov to Lyapunov mission, we first
compute an admissible trajectory using a heteroclinic orbit between the two
periodic orbits. It is then used to initialize a multiple shooting method in
order to release the constraint. We finally optimize the terminal points on the
periodic orbits. Moreover, we use continuation methods on position and on
thrust, in order to gain robustness. A more general Halo to Halo mission, with
different energies, is computed in the last section without heteroclinic orbits
but using invariant manifolds to initialize shooting methods with a similar
approach
Application of optimal control theory in finance and economy
The aim of current master thesis is to give the appropriate knowledge for the full understanding of models used in the optimization of the economical processes. A comparison was made of whether the size of the company influences the order of the solution and its general look. Now it’s known that both huge and tiny companies, as well as individuals, who are about to make some investment decision, and use optimal control theory for the optimization of their activity. The model of the optimal economic growth can easily find its use in real economic and experience various improvements and extensions. There might be derived the unified models for groups of typical cases, as we can say that all decisions to be made can be summed under one variable
Transversality Conditions and Dynamic Economic Behavior
Transversality conditions are optimality conditions often used along with Euler equations to characterize the optimal paths of dynamic economic models. This article explains the foundations of transversality conditions using a geometric example, a finite horizon problem, and an infinite horizon problem. Their relationships to asset bubbles, hyperdeflations, and no-Ponzi-game conditions are also discussed.
The Variable-Order Fractional Calculus of Variations
This book intends to deepen the study of the fractional calculus, giving
special emphasis to variable-order operators. It is organized in two parts, as
follows. In the first part, we review the basic concepts of fractional calculus
(Chapter 1) and of the fractional calculus of variations (Chapter 2). In
Chapter 1, we start with a brief overview about fractional calculus and an
introduction to the theory of some special functions in fractional calculus.
Then, we recall several fractional operators (integrals and derivatives)
definitions and some properties of the considered fractional derivatives and
integrals are introduced. In the end of this chapter, we review integration by
parts formulas for different operators. Chapter 2 presents a short introduction
to the classical calculus of variations and review different variational
problems, like the isoperimetric problems or problems with variable endpoints.
In the end of this chapter, we introduce the theory of the fractional calculus
of variations and some fractional variational problems with variable-order. In
the second part, we systematize some new recent results on variable-order
fractional calculus of (Tavares, Almeida and Torres, 2015, 2016, 2017, 2018).
In Chapter 3, considering three types of fractional Caputo derivatives of
variable-order, we present new approximation formulas for those fractional
derivatives and prove upper bound formulas for the errors. In Chapter 4, we
introduce the combined Caputo fractional derivative of variable-order and
corresponding higher-order operators. Some properties are also given. Then, we
prove fractional Euler-Lagrange equations for several types of fractional
problems of the calculus of variations, with or without constraints.Comment: The final authenticated version of this preprint is available online
as a SpringerBrief in Applied Sciences and Technology at
[https://doi.org/10.1007/978-3-319-94006-9]. In this version some typos,
detected by the authors while reading the galley proofs, were corrected,
SpringerBriefs in Applied Sciences and Technology, Springer, Cham, 201
Non-constant discounting in finite horizon: The free terminal time case
This paper derives the HJB (Hamilton-Jacobi-Bellman) equation for sophisticated agents in a finite horizon dynamic optimization problem with non-constant discounting in a continuous setting, by using a dynamic programming approach. A simple example is used in order to illustrate the applicability of this HJB equation, by suggesting a method for constructing the subgame perfect equilibrium solution to the problem. Conditions for the observational equivalence with an associated problem with constant discounting are analyzed. Special attention is paid to the case of free terminal time. Strotzs model (an eating cake problem of a nonrenewable resource with non-constant discounting) is revisited.naive and sophisticated agents, observational equivalence, non-constant discounting, free terminal time
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