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2013 | Buch

Functional Analysis, Calculus of Variations and Optimal Control

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Functional analysis owes much of its early impetus to problems that arise in the calculus of variations. In turn, the methods developed there have been applied to optimal control, an area that also requires new tools, such as nonsmooth analysis. This self-contained textbook gives a complete course on all these topics. It is written by a leading specialist who is also a noted expositor.

This book provides a thorough introduction to functional analysis and includes many novel elements as well as the standard topics. A short course on nonsmooth analysis and geometry completes the first half of the book whilst the second half concerns the calculus of variations and optimal control. The author provides a comprehensive course on these subjects, from their inception through to the present. A notable feature is the inclusion of recent, unifying developments on regularity, multiplier rules, and the Pontryagin maximum principle, which appear here for the first time in a textbook. Other major themes include existence and Hamilton-Jacobi methods.

The many substantial examples, and the more than three hundred exercises, treat such topics as viscosity solutions, nonsmooth Lagrangians, the logarithmic Sobolev inequality, periodic trajectories, and systems theory. They also touch lightly upon several fields of application: mechanics, economics, resources, finance, control engineering.

Functional Analysis, Calculus of Variations and Optimal Control is intended to support several different courses at the first-year or second-year graduate level, on functional analysis, on the calculus of variations and optimal control, or on some combination. For this reason, it has been organized with customization in mind. The text also has considerable value as a reference. Besides its advanced results in the calculus of variations and optimal control, its polished presentation of certain other topics (for example convex analysis, measurable selections, metric regularity, and nonsmooth analysis) will be appreciated by researchers in these and related fields.

Inhaltsverzeichnis

Frontmatter

Functional Analysis

Frontmatter
Chapter 1. Normed Spaces
Abstract
We now set off on an expedition through the vast subject of functional analysis. No doubt the reader has some familiarity with this place, and will recognize some of the early landmarks of the journey. Our starting point is the study of normed spaces, which are situated at the confluence of two far-reaching mathematical abstractions: vector spaces, and topology. The setting is that of a vector space over the real numbers \({\mathbb{R}}\). The central idea of this chapter is that of a norm. It is studied in detail, along with the attendant concept of linear operators. The dual space is introduced. The chapter ends with a presentation of derivates, directional derivatives, tangent and normal vectors. These constructs, which allow one to reduce nonlinear situations to linear ones, will play a central role in later developments.
Francis Clarke
Chapter 2. Convex sets and functions
Abstract
The class of convex sets plays a central role in the theory. The subset C of the normed space X is said to be convex if
$$t\in\, (0 , 1)\,,\:\:\: x , y\: \in \:C\:\: \Longrightarrow\:\: (1-t) x+ t y\: \in \:C\,.$$
The separation of convex sets is the issue treated by the celebrated Hahn-Banach theorem, often said to be the most basic tool of classical functional analysis. We also study convex functions, a more modern consideration. Such functions, which turn out to have special regularity properties, will play a big role in things to come. We also introduce lower semicontinuous and extended-valued functions, which are important later in optimization. Support functions and indicator functions are important examples of these.
Francis Clarke
Chapter 3. Weak topologies
Abstract
The existence of a minimum for an optimization problem such as minA f is a sensitive issue in an infinite dimensional space X, since the compactness of A may be difficult to secure. The compactness fails, notably, when A is the unit ball, and it is quite possible that a continuous linear functional may not attain a minimum over B X . What if we were to change the topology on X, in an attempt to mitigate this lack of compactness? We would want to have fewer open sets: this is called weakening the topology. The reason behind this is simple: the fewer are the open sets in a topology, the more likely it becomes that a given set is compact. On the other hand, the fewer open sets there are, the harder it is for a function defined on the space to be continuous (or lower semicontinuous), which is the other main factor in guaranteeing existence. The tension between these two contradictory pressures, the problem of finding the “right” topology that establishes a useful balance between them, is one of the great themes of functional analysis, and the subject of this chapter.
Francis Clarke
Chapter 4. Convex analysis
Abstract
The reader has been told that convex functions are of particular interest. One reason for this (among others) lies in the fact that a certain generalized calculus can be developed for convex functions (and sets), one that is often referred to nowadays as convex analysis. The subject has widespread applications. The role of the derivative in convex analysis is played by the subdifferential, the collection of subgradients. An element ζ of X is a subgradient of f at x (in the sense of convex analysis) if it satisfies the inequality
$$f(y) - f(x) \: \geqslant \:\langle \, \zeta ,\,y-x \,\rangle ~\; \forall \,y\in\, X.$$
The operation of conjugacy is also central to convex analysis. Before examining convex calculus however, we add to our toolkit a useful result on permuting min and max, a result called the minimax theorem.
Francis Clarke
Chapter 5. Banach spaces
Abstract
A normed space X is said to be a Banach space if its metric topology is complete. This means that every Cauchy sequence x i in X admits a limit in X: there exists a point x∈ X such that ∥ x i x ∥→ 0. Informally, the reader may understand the absence of such a point x as meaning that the space has a hole where x should be. For purposes of minimization, one of our principal themes, it is clear that the existence of minimizers is imperiled by such voids. The existence of solutions to minimization problems is not the only compelling reason, however, to require the completeness of a normed space, as we shall see. The property is essential in making available to us certain basic tools developed in this chapter, such as uniform boundedness, minimization principles, and weak compactness.
Francis Clarke
Chapter 6. Lebesgue spaces
Abstract
The Lebesgue spaces L p (Ω) play a central role in many applications of functional analysis. This chapter focuses upon their basic properties, as well as certain attendant issues that will be important later. Notable among these are the semicontinuity of integral functionals, and the existence of measurable selections. We begin by identifying a geometric property of the norm which, when present, turns out to have a surprising consequence. A normed space X is said to be uniformly convex if it satisfies the following property:
$$\forall ~ \varepsilon > 0\,, ~ \exists ~ \delta\, >\, 0 \mbox{ such that } x \in\, B\,,\: y \in\, B\,, \,\: \|\, x - y\, \| >\, \varepsilon \:\: \Longrightarrow \:\: \Big\| \, \frac {x + y}{2}\, \Big\| ~ <\, 1 - \delta\,.$$
In geometric terms, this is a way of saying that the unit ball is curved. The property depends upon the choice of the norm on X, even among equivalent norms, as one can see even in \({\mathbb{R}}^{\,2}\). Yet it implies an intrinsic property independent of the choice of norm: a uniformly convex Banach space is reflexive.
Francis Clarke
Chapter 7. Hilbert spaces
Abstract
A Banach space X is called a Hilbert space if it admits a bilinear mapping (x,y) ↦ 〈 x ,y 〉 X that generates its norm, in the following sense:
$$\|\,x\,\|^{\,2} \: = \:\langle \, x\,, x \,\rangle _X\;~\; \forall \,x\in\, X .$$
The bilinear mapping is then referred to as an inner product on X. Canonical cases of Hilbert spaces include \({\mathbb{R}}^{ n} \), L 2(Ω), and  2. Some rather remarkable consequences for the structure of the space X follow from the mere existence of this scalar product. We begin the chapter with a review of the basic theory of Hilbert spaces. Then we present a smooth minimization principle, in which the differentiability of the Hilbert norm plays a crucial role. The final sections introduce the proximal subdifferential, an important tool in nonsmooth (and nonconvex) analysis.
Francis Clarke
Chapter 8. Additional exercises for Part I
Abstract
The chapter consists of several dozen exercises on functional analysis, the first three of which are the following:
  • Give an example of a locally Lipschitz function \(f:X\to\,{\mathbb{R}}\) defined on a Hilbert space X which is not bounded below on the unit ball. Could such a function be convex?
  • Let A be a bounded subset of a normed space X. Prove that \({\text{co}}\,\big(\partial A\big)\,\supset\, { \text{cl}\,}\, A\,. \)
  • Let X be a normed space, and let A be an open subset having the property that each boundary point x of A admits a supporting hyperplane; that is, there exist 0 ≠ ζ x ∈ X and \(c_{x}\in\,{\mathbb{R}}\) such that
    $$\langle \, \zeta_{\,x}\,, x\,\rangle \: = \:c_x\:,\;\; \langle \, \zeta_{\,x}\,, u\,\rangle \: \leqslant \:\, c_x~\; \forall \,u\in\, A\,. $$
    Prove that A is convex. Prove that the result remains valid if the hypothesis “A is open” is replaced by “A is closed and has nonempty interior.”
Francis Clarke

Optimization and Nonsmooth Analysis

Frontmatter
Chapter 9. Optimization and multipliers
Abstract
The abstract optimization problem minA f consists of minimizing a cost function f(x) over the points x belonging to the admissible set A. The nature of A, and also of the function f, determine whether our problem is classical or modern, discrete or continuous, finite or infinite dimensional, smooth or convex. Optimization is a rich and varied subject with numerous applications. The core mathematical issues, however, are always the same:
  • Existence: Is there, in fact, a solution of the problem?
  • Necessary conditions: What special properties must a solution have, properties that will help us to identify it?
  • Sufficient conditions: Having identified a point that is suspected of being a solution, what tools can we apply to confirm the suspicion?
We study these issues, paying particular attention to the distinction between the deductive and inductive methods, and to the method of multipliers.
Francis Clarke
Chapter 10. Generalized gradients
Abstract
Let \(f:X\to\,{\mathbb{R}}\) be a lower semicontinuous function defined on a Banach space X. If f is continuously differentiable, then the mean value theorem implies that it is locally Lipschitz. If, instead, the function f is convex, then once more it has the property of being locally Lipschitz, as we have seen. In this sense, the class of locally Lipschitz functions subsumes the smooth and convex cases. The class of locally Lipschitz functions has other features that recommend it as an environment in which to develop a theory of nonsmooth calculus, which is our goal in this chapter. It is closed under familiar operations such as sum, product, and composition. But it is also closed under less classical ones, such as taking lower or upper envelopes. Finally, it includes certain nonsmooth, nonconvex functions that are important in a variety of applications, notably distance functions. We proceed to develop the calculus of the generalized gradient of a locally Lipschitz function f, denoted C f(x). This leads to a unified treatment of smooth and convex calculus, as well as an associated geometric theory of tangents and normals to arbitrary closed sets.
Francis Clarke
Chapter 11. Proximal analysis
Abstract
We proceed in this chapter to develop the calculus (and the geometry) associated with the proximal subdifferential. The reader has encountered this object in Chapter 7, in the context of Hilbert spaces. Let \(f:{\mathbb{R}}^{ n}\to {\mathbb{R}}_{ \infty}\) be given, and let x∈ dom f. Recall that \(\zeta\in\,{\mathbb{R}}^{ n}\) is a proximal subgradient of f at x if for some σ=σ(x,ζ) ⩾ 0, and for some neighborhood V=V(x,ζ) of x, we have
$$f(y)-f(x)+\sigma|\,y-x\,|^{\,2}\:\geqslant\: \langle \,\zeta,\, y-x\,\rangle ~\; \forall \,y\in\, V. $$
This is referred to as the proximal subgradient inequality. The collection of all such ζ (which may be empty) is the proximal subdifferential of f at x, denoted P f(x). The cornerstone of our development of proximal calculus is a multi-directional extension of the mean value theorem. We also study the related theory of proximal normals, establish a multiplier rule in proximal terms, and explain the connection between viscosity (or Dini) subgradients and the other generalized derivatives that we have encountered.
Francis Clarke
Chapter 12. Invariance and monotonicity
Abstract
A venerable notion from the classical theory of dynamical systems is that of flow invariance. When the basic model consists of an autonomous ordinary differential equation x ′(t)=f(x(t)) and a set S, then flow invariance of the pair (S,f) is the property that for every initial point α∈ S, the solution x(⋅) satisfying x(0)=α remains in S : x(t)∈ S for all t ⩾ 0. In this chapter, we study highly useful generalizations of this concept to situations wherein the differential equation is replaced by an inclusion. A trajectory x of the multifunction F, on a given interval [ a,b ], refers to a function \(x:[\,a ,b\,]\to\,{\mathbb{R}}^{ n}\) which satisfies the differential inclusion
$$x \,' (t)\: \in \:F( x(t)) , \;\;t\in\, [\,a ,b\,]{~\,\text{a.e.}}$$
When F(x) is a singleton {f(x)} for each x, the differential inclusion reduces to an ordinary differential equation. Otherwise, we would generally expect there to be multiple trajectories from the same initial condition. In such a context, the invariance question bifurcates: do we require some of, or all of, the trajectories to remain in S? This will be the difference between weak and strong invariance.
Francis Clarke
Chapter 13. Additional exercises for Part II
Abstract
This chapter consists of a collection of problems bearing upon optimization and nonsmooth analysis, the first of which is the following: Let M be an n×n symmetric matrix. Consider the optimization problem
$$ \text{Minimize }\: \langle\, x , M x \,\rangle\:\:\: \text{subject to}\:\: \: h(x)\,:=\, 1-|\,x\,|^{\,2}\: =\:0 ,\:\;x\in\, {\mathbb{R}}^{ n}. $$
(a)
Observe that a solution x exists, and write the conclusions of the multiplier rule for this problem. Show that they cannot hold abnormally. It follows that the multiplier in this case is of the form (1,λ).
 
(b)
Deduce that λ is an eigenvalue of M, and that λ =〈 x ,Mx  〉.
 
(c)
Prove that λ coincides with the first, or least eigenvalue λ  1 of M (this statement makes sense because the eigenvalues are real). Deduce the Rayleigh formula, which asserts that λ  1 is given by min {〈 x,Mx 〉:| x |=1 }.
 
Francis Clarke

Calculus of Variations

Frontmatter
Chapter 14. The classical theory
Abstract
The basic problem in the subject that is referred to as the calculus of variations consists in minimizing an integral functional of the type
$$J(x) \: =\: \int_{a}^{\,b} \Lambda \big(t,\, x(t),\, x \,' (t)\big)\,dt $$
over a class of functions x defined on the interval [ a,b ], and which take prescribed values at a and b. The study of this problem (and its numerous variants) is over three centuries old, yet its interest has not waned. Its applications are numerous in geometry and differential equations, in mechanics and physics, and in areas as diverse as engineering, medicine, economics, and renewable resources. It is not surprising, then, that modeling and numerical analysis play a large role in the subject today. In the following chapters, however, we present a course in the calculus of variations which focuses on the core mathematical issues: necessary conditions, sufficient conditions, existence theory, regularity of solutions. This chapter deals with the case in which these variables are one-dimensional and all the data are smooth.
Francis Clarke
Chapter 15. Nonsmooth extremals
Abstract
Is it always satisfactory to consider only smooth solutions of the basic problem, as we have done in the previous chapter? By the middle of the 19th century, the need to go beyond continuously differentiable functions was becoming increasingly apparent. In fact, the calculus of variations may have been the first subject to acknowledge a need to consider nonsmooth functions. We begin by exhibiting a simple problem that has a natural solution in the class of piecewise-smooth functions, but has no solution in that of smooth functions. This shows that the very existence of solutions is an impetus for admitting nonsmooth arcs. The need also became apparent in physical applications (soap bubbles, for example, generally have corners and creases). Spurred by these considerations, the theory of the basic problem was extended to the context of piecewise-smooth functions. In this chapter, we develop this theory, but within the more general class of Lipschitz functions x. We extend the setting of the basic problem in one more way, by allowing x to be vector-valued.
Francis Clarke
Chapter 16. Absolutely continuous solutions
Abstract
The theory of the calculus of variations at the turn of the twentieth century lacked a critical component: it had no existence theorems. These constitute an essential ingredient of the deductive method, the approach whereby one combines existence, rigorous necessary conditions, and examination of candidates to arrive at a solution. When it applies, it often leads to the conclusion that a global minimum exists, whereas the classical methods assert only the existence of a local minimum. In mechanics, a local minimum is a meaningful goal, since it generally corresponds to a stable configuration of the system. In many modern applications however (such as in engineering or economics), only global minima are of real interest. Along with the quest for the multiplier rule (which we discuss in the next chapter), it was the longstanding question of existence that dominated the scene in the calculus of variations in the first half of the twentieth century. The key step in developing existence theory is to extend the context of the basic problem to admit absolutely continuous functions.
Francis Clarke
Chapter 17. The multiplier rule
Abstract
The reader has been told that the great twentieth-century quests in the calculus of variations have involved existence and multiplier rules. Progress in functional analysis, together with the direct method, has largely resolved the former issue; we turn now to that of multipliers. We consider the classical problem of Lagrange for this purpose. It consists of the basic problem (P) to which has been grafted an additional pointwise equality constraint φ(t,x,x ′) = 0. The additional constraint makes this problem much more complex than (P), or even the isoperimetric problem. In part, this is because we now have infinitely many constraints, one for each t. Given our experience in optimization, we expect the multiplier rule to assert, in a now familiar pattern, that if x solves this problem, then there exist multipliers η,λ, not both zero, with η= 0 or 1, such that x satisfies the necessary conditions for the Lagrangian ηΛ+λ(t)φ. Note that λ is a function of t here, which is to be expected, since there is a constraint φ(t,x,x ′) = 0 for each t.
Francis Clarke
Chapter 18. Nonsmooth Lagrangians
Abstract
We now take a step in a direction never envisaged by the classical theory: the introduction of nonsmooth Lagrangians (as opposed to nonsmooth solutions). This is a modern issue that stems from new applications in such disciplines as engineering, economics, mechanics, and operations research. At the same time, this factor will play an essential role in the proof of such theorems as the multiplier rules of the preceding chapter, even when the underlying problems have smooth data. The new necessary conditions differ most from the earlier ones in the Euler equation, which now becomes an inclusion:
$$ p\,' (t)\,\in\, {\text {co}}\,\big\{ \, \omega: \big(\omega , p(t)\big)\in\, \partial_{ L}\,\Lambda \big(t, x_{ *}(t), x_* '(t)\big)\big\} {~\,\text {a.e.}}\; t\in\, [\,a ,b\,] . $$
We give examples to show why it can be useful to have weakened the regularity hypotheses on the Lagrangian, as is done here.
Francis Clarke
Chapter 19. Hamilton-Jacobi methods
Abstract
In the preceding chapters, the predominant issues have been those connected with the deductive method: existence on the one hand, and the twin issues of regularity and necessary conditions on the other. We now introduce the reader to verification functions, a technique which unifies all the main inductive methods. It will be seen that this leads to a complex of ideas centered around the Hamilton-Jacobi inequality (or equation). To illustrate the method, we give an elementary proof of the celebrated logarithmic Sobolev inequality. The final section considers the following Cauchy problem for the Hamilton-Jacobi equation:
$$\mathbf{(H J)}\quad \left\{\quad \begin{aligned} u_{\,t} + H( x , u_{\, x}) &\: = \:0 ,\:\: (t, x)\,\in\, \Omega\,:=\, (0 ,\infty) \times {\mathbb{R}}^{ n}\\ u(0 , x) &\: = \:\ell(x)\,,\;\; x\in\, {\mathbb{R}}^{ n}. \end{aligned} \right. $$
The need to consider generalized solution concepts is explained, and the connection to viscosity solutions is made.
Francis Clarke
Chapter 20. Multiple integrals
Abstract
The problem under consideration is the minimization of the functional
$$\quad J(u)\:\, =\:\: \int_\Omega F\big( x , u(x),\,D u(x)\big)\, dx , $$
over a class X of real-valued functions u defined on Ω, where Ω denotes a nonempty open bounded subset of \({\mathbb{R}}^{ n}\). The values of the admissible functions are prescribed on the boundary of Ω :
$$u(x)\: = \:\varphi(x)~\; \forall \,x\: \in \:\Gamma\, :=\:\, \partial\Omega\,, $$
where φ is given. Of course, the reader will recognize this as an extension to multiple integrals of the basic problem in the calculus of variations; this chapter serves as an introduction to that topic. As in the single integral case, it is perhaps the choice of underlying space X that makes the greatest difference in the ensuing theory.
Francis Clarke
Chapter 21. Additional exercises for Part III
Abstract
The chapter consists of several dozen exercises in the calculus of variations, the first of which is the following: Consider the problem
$$\min\;\;\; \int_{ 1}^{\,3} \big\{ \, t \big( x\,' (t)\big)^{2}-x(t)\big\} \, dt\: :\: x\in\, C^{\,2}[\,1,3\,] ,\;\; x(1)\,=\, 0 ,\; \: x(3)\,=-1. $$
(a)
Find the unique admissible extremal x .
 
(b)
Prove that x is a global minimizer for the problem.
 
(c)
Prove that the problem
$$\min\;\; J(x)\:=\: \int_{-2}^{\,3} \big\{ \, t \big( x\,' (t)\big)^{2}-x(t)\big\} \, dt\: :\: x\in\, C^{\,2}[-2\,,3\,] ,\; x(-2)\,=\, A\,,\; \: x(3)\,=B $$
admits no local minimizer, regardless of the values of A and B.
 
Francis Clarke

Optimal Control

Frontmatter
Chapter 22. Necessary conditions
Abstract
Systems of ordinary differential equations such as x ′(t) = f(t,x(t)) are routinely used today to model a wide range of phenomena, in areas as diverse as aeronautics, power generation, robotics, economic growth, and natural resources. The great success of this paradigm is due in part to the fact that it suggests a natural mechanism through which the behavior of the system can be influenced by external factors. This is done by introducing an explicit control variable in the differential equation, a time-varying parameter that can be chosen (within certain limits) so as to attain a certain goal. This leads to the controlled differential equation
$$ x\,' (t) \: = \:f( t, x(t) , u(t)) ,\,\:\: u(t)\in\, U. $$
We present in this chapter the celebrated Pontryagin maximum principle, a set of necessary conditions for the optimal control of such a system.
Francis Clarke
Chapter 23. Existence and regularity
Abstract
To this point, our discussion of optimal control has focused on the issue of necessary conditions. Before turning to the attendant issues of existence and regularity, we digress somewhat to take note of an entirely new consideration, one that is called relaxation. Just as an unstable equilibrium in mechanics is considered somewhat meaningless, one may consider that certain optimal control problems are not well posed, since their solution lacks stability, in a certain sense. It is the goal of relaxation to reformulate the problem so as to avoid this phenomenon. We then proceed to obtain three existence theorems for optimal control. Roughly speaking, these theorems require that velocity sets be convex. When there is a running cost, its convexity relative to the control variable is the functional counterpart of that property. In addition, some growth restriction is required. This is most easily supplied by taking the control set to be compact; otherwise, when the controls are unbounded, coercivity of the running cost can be postulated. We close the chapter by identifying criteria under which the optimal control must be continuous.
Francis Clarke
Chapter 24. Inductive methods
Abstract
Sometimes we suspect that we have identified a solution of an optimal control problem, but the deductive reasoning that would allow us to assert its optimality is unavailable. This might happen because no existence theorem applies, or because the applicability of necessary conditions is uncertain. In this situation, we may seek to use an inductive method to confirm the optimality of the suspect. We describe three such methods in this chapter, the first of which is based on a strengthening of the conditions that appear in the maximum principle. We show that, in the convex case of the problem (properly interpreted), the maximum principle (in normal form, and somewhat strengthened) is indeed a sufficient condition for optimality. The next method considered is that of verification functions, studied in detail in the calculus of variations. We show that it carries over to the optimal control setting. Finally, we illustrate in the last section how a uniqueness theorem for generalized solutions of the Hamilton-Jacobi equation leads to an inductive method allowing us to confirm conjectured optimality.
Francis Clarke
Chapter 25. Differential inclusions
Abstract
We develop necessary conditions for the optimal control of systems governed by a differential inclusion, in the context of the following problem:
$$ \left\{ \begin{array}{ll} \quad \text{Minimize } \quad &J(x)\: =\: \displaystyle{\ell\big( x(a) , x(b)\big)} \\ \quad \text{subject to} &x \,' (t)\: \in \:F_{\,t}\big( x(t)\big) ,\;\; t\in\, [\,a ,b\,] {~\,\text {a.e.}}\\ &\big( x(a) , x(b)\big)\in\, E . \end{array} \right. $$
This differential inclusion problem seems far less natural than the standard control formulation, and it does not lend itself very well to modeling. Nonetheless, it provides an ideal mathematical environment in which to prove various types of necessary conditions. All the different versions of the maximum principle presented so far have followed from the extended maximum principle; this, in turn, will be a consequence of the theorems proved in this chapter.
Francis Clarke
Chapter 26. Additional exercises for Part IV
Abstract
This chapter consists of a collection of exercises on optimal control, the first of which is the following: Optimal control problems which arise in economics or finance often feature a discount rate δ. This parameter is used to express the present value of future revenues or expenses. We introduce this consideration in the problem described on p. 445, by modifying the cost functional (but nothing else) as follows:
$$J(x ,u)\: = \:\int_{ 0}^{\,T} e^{-\delta t}\big(\, x(t)+u(t)\big)\, dt\,. $$
For small δ>0, it can be shown that the optimal process has the same general turnpike nature as before, but with a different turnpike value (depending on δ) for the state. Determine that value.
Francis Clarke
Backmatter
Metadaten
Titel
Functional Analysis, Calculus of Variations and Optimal Control
verfasst von
Francis Clarke
Copyright-Jahr
2013
Verlag
Springer London
Electronic ISBN
978-1-4471-4820-3
Print ISBN
978-1-4471-4819-7
DOI
https://doi.org/10.1007/978-1-4471-4820-3