# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

April 2011 , Volume 29 , Issue 2

Special Issue on Control, Nonsmooth Analysis and Optimization
Celebrating the 60th Birthday of Francis Clarke and Richard Vinter

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2011, 29(2): i-ii doi: 10.3934/dcds.2011.29.2i +[Abstract](1449) +[PDF](6552.7KB)
Abstract:
These preface words, quoted in Francis Clarke's 1983 book, stroke a sensitive chord and set the tone for a vision fueling the construction of an extremely rich body of intertwined developments in nonsmooth analysis, optimization and control by a carefully networked community.
This special issue comprises post-conference articles from a selection of works presented at the Workshop on Control, Nonsmooth Analysis and Optimization, held in Porto, Portugal in May 2009. The workshop was part of the celebrations of the 60th birthday of Francis Clarke and Richard B. Vinter.

2011, 29(2): 403-415 doi: 10.3934/dcds.2011.29.403 +[Abstract](1723) +[PDF](381.3KB)
Abstract:
This paper concerns the investigation of a general impulsive control problem. The considered impulsive processes are of non-standard type: control processes admit ordinary type controls as the impulse develops. New necessary conditions of optimality in the form of Pontryagin Maximum Principle are obtained. These conditions are applied to a model problem and are shown to yield useful information about optimal control modes.
2011, 29(2): 417-437 doi: 10.3934/dcds.2011.29.417 +[Abstract](2987) +[PDF](509.9KB)
Abstract:
We introduce a discrete-time fractional calculus of variations. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler--Lagrange and Legendre type conditions are given. They show that the solutions of the fractional problems coincide with the solutions of the corresponding non-fractional variational problems when the order of the discrete derivatives is an integer value.
2011, 29(2): 439-451 doi: 10.3934/dcds.2011.29.439 +[Abstract](1471) +[PDF](611.0KB)
Abstract:
We prove existence of minimizers for the multiple integral

$\int$Ω$\l(u(x),\rho_1(x,u(x))$∇$u(x))\ \ \rho_2(x,u(x))\ dx \ \ \ on\ \ \$W1,1u(Ω),(*)

where Ω$\subset\R^d$ is open bounded, $u:$Ω$\toR$ is in the Sobolev space u($*$)+W1,10(Ω), with boundary data $u_$$(\cdot)\inW1,1(Ω)\cap C^{0}(Ω); and \l:RΧR^d\to[0,\infty] is superlinear L\oxB-measurable with \rho_1(\cdot,\cdot),\rho_2(\cdot,\cdot)\in C^{0}(ΩΧR) both >0. One main feature of our result is the unusually weak assumption on the lagrangian: l**(\cdot,\cdot) only has to be lsc at (\cdot,0), i.e. at zero gradient. Here l**(s,\cdot) denotes the convex-closed hull of \l(s,\cdot). We also treat the nonconvex case \l(\cdot,\cdot)\nel**(\cdot,\cdot), whenever a well-behaved relaxed minimizer is a priori known. Another main feature is that \l(s,\xi)=\infty is freely allowed, even at zero gradient, so that (*) may be seen as the variational reformulation of optimal control problems involving implicit first-order nonsmooth scalar partial differential inclusions under state and gradient pointwise constraints. The general case \intΩL(x,u(x),∇u(x)) is also treated, though with less natural hypotheses, but still allowing L(x,\cdot,\xi) non-lsc for \xi\ne0. 2011, 29(2): 453-466 doi: 10.3934/dcds.2011.29.453 +[Abstract](1657) +[PDF](347.7KB) Abstract: We provide intrinsic sufficient conditions on a multifunction F and endpoint data φ so that the value function associated to the Mayer problem is semiconcave. 2011, 29(2): 467-484 doi: 10.3934/dcds.2011.29.467 +[Abstract](1639) +[PDF](438.1KB) Abstract: We prove validity of the classical DuBois-Reymond differential inclusion for the minimizers y(\cdot) of the integral \int_{a}^{b}L( x( t) ,x^'( t)) d\,t,\text{ \ }x\( \cdot) \in W^{1,1}((a,b) ,\mathbb{R}^{n}) ,\text{ \ }x(a)=A\,x(b) =B\ \ (*) whose velocities are not a.e. constrained by the domain boundary. Thus we do not ask ( as preceding results do) the free-velocity times T_{f ree}:=\{ t\in[ a,b] :y^'( t) \in int\text{ }dom\ L( y\( t) ,\cdot) \} to have "full measure"; on the contrary, "positive measure" of T_{f ree} suffices here to guarantee the above necessary condition. One main feature of our result is that L( S,\xi) =\infty freely allowed, hence the domains dom$$L( S,\cdot)$ may be e.g. compact and (*) can be seen as the variational reformulation of general state-and-velocity constrained optimal control problems.
Another main feature is the clean generality of our assumptions on $L( \cdot) :$ any Borel-measurable function $L:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow[ 0,\infty]$ having $L( \cdot,0)$ $lsc$ and $L( S,\cdot)$ convex $lsc$ $\forall\,S.$
The nonconvex case is also considered, for $L( S,\cdot)$ almost convex lsc $\forall\,S.$

2011, 29(2): 485-503 doi: 10.3934/dcds.2011.29.485 +[Abstract](1763) +[PDF](433.6KB)
Abstract:
We give a relatively short and self-contained proof of a theorem that asserts necessary conditions for a general optimal control problem. It has been shown that this theorem, which is simple to state, provides a powerful template from which necessary conditions for various other problems in dynamic optimization can be directly derived, at the level of the state of the art. These include various extensions of the Pontryagin maximum principle and the multiplier rule.
2011, 29(2): 505-522 doi: 10.3934/dcds.2011.29.505 +[Abstract](1577) +[PDF](374.1KB)
Abstract:
In this paper we report conditions ensuring Lipschitz continuity of optimal control and Lagrange multipliers for a dynamic optimization problem with inequality pure state and mixed state-control constraints.
2011, 29(2): 523-545 doi: 10.3934/dcds.2011.29.523 +[Abstract](1457) +[PDF](400.2KB)
Abstract:
We consider a general optimal control problem with intermediate and mixed constraints. Using a natural transformation (replication of the state and control variables), this problem is reduced to a standard optimal control problem with mixed constraints, which makes it possible to obtain quadratic order conditions for an "extended" weak minimum. The conditions obtained are applied to the problem of light refraction.
2011, 29(2): 547-557 doi: 10.3934/dcds.2011.29.547 +[Abstract](1425) +[PDF](337.1KB)
Abstract:
We consider autonomous, second order problems in the calculus of variations in one independent variable. For analogous first order problems it is known that, under standard hypotheses of existence theory and a local boundedness condition on the Lagrangian, minimizers over $W^{1,1}$ have bounded first derivatives ($W^{1,\infty}$ regularity prevails). For second order problems one might expect, by analogy, that minimizers would have bounded second derivatives ($W^{2,\infty}$ regularity) under the standard existence hypotheses $(HE)$ for second order problems, supplemented by a local boundedness condition. A counter-example, however, indicates that this is not the case. In earlier work, $W^{2, \infty}$ regularity has been established for these problems under $(HE)$ and additional 'integrability' hypotheses on derivatives of the Lagrangian, evaluated along the minimizer. We show that these additional hypotheses can be significantly reduced. The proof techniques employed depend on a combination of the application of a change of independent variable and of extensions to Tonelli regularity theory proved by Clarke and Vinter.
2011, 29(2): 559-575 doi: 10.3934/dcds.2011.29.559 +[Abstract](2050) +[PDF](723.3KB)
Abstract:
We address necessary conditions of optimality (NCO), in the form of a maximum principle, for optimal control problems with state constraints. In particular, we are interested in the NCO that are strengthened to avoid the degeneracy phenomenon that occurs when the trajectory hits the boundary of the state constraint. In the literature on this subject, we can distinguish two types of constraint qualifications (CQ) under which the strengthened NCO can be applied: CQ involving the optimal control and CQ not involving it. Each one of these types of CQ has its own merits. The CQs involving the optimal control are not so easy to verify, but, are typically applicable to problems with less regularity on the data. In this article, we provide conditions under which the type of CQ involving the optimal control can be reduced to the other type. In this way, we also provide nondegenerate NCO that are valid under a different set of hypotheses.
2011, 29(2): 577-593 doi: 10.3934/dcds.2011.29.577 +[Abstract](1790) +[PDF](429.3KB)
Abstract:
In this paper we consider the problem of the calculus of variations for a functional which is the composition of a certain scalar function $H$ with the delta integral of a vector valued field $f$, i.e., of the form $H (\int_{a}^{b}f(t,x^{\sigma}(t),x^{\Delta}(t))\Delta t)$. Euler-Lagrange equations, natural boundary conditions for such problems as well as a necessary optimality condition for isoperimetric problems, on a general time scale, are given. A number of corollaries are obtained, and several examples illustrating the new results are discussed in detail.
2011, 29(2): 595-613 doi: 10.3934/dcds.2011.29.595 +[Abstract](1383) +[PDF](446.9KB)
Abstract:
We develop a notion of generalized solution to a stochastic differential equation depending in a nonlinear way on a vector--valued stochastic control process $\{U_t\},$ merely of bounded variation, and on its derivative. Our results rely on the concept of Lipschitz continuous graph completion of $\{U_t\}$ and the generalized solution turns out to coincide a.e. with the limit of classical solutions to (1). In the linear case our notion of solution is equivalent to the usual one in distributional sense. We prove that the generalized solution does not depend on the particular graph-completion of the control process $\{U_t\}$ both for vector-valued controls under a suitable commutativity condition and for scalar controls.
2011, 29(2): 615-622 doi: 10.3934/dcds.2011.29.615 +[Abstract](1468) +[PDF](487.9KB)
Abstract:
We consider sets $S\subset\R^n$ satisfying a certain exterior sphere condition, and it is shown that under wedgedness of $S$, it coincides with $\varphi$-convexity. We also offer related improvements concerning the union of uniform closed balls conjecture.
2011, 29(2): 623-646 doi: 10.3934/dcds.2011.29.623 +[Abstract](1650) +[PDF](492.6KB)
Abstract:
The notions of $V$-Jacobian and $V$-co-Jacobian are introduced for locally Lipschitzian functions acting between arbitrary normed spaces $X$ and $Y$, where $V$ is a subspace of the dual space $Y^*$. The main results of this paper provide a characterization, calculus rules and also the computation of these Jacobians of piecewise smooth functions.
2011, 29(2): 647-670 doi: 10.3934/dcds.2011.29.647 +[Abstract](1713) +[PDF](431.8KB)
Abstract:
The paper deals with optimal control problems described by higher index DAEs. We introduce a numerical procedure for solving these problems. The procedure, based on the appropriately defined adjoint equations, refers to an implicit Runge--Kutta method for differential--algebraic equations. Assuming that higher index DAEs can be solved numerically the gradients of functionals defining the control problem are evaluated with the help of well--defined adjoint equations. The paper presents numerical examples related to index three DAEs showing the validity of the proposed approach.
2011, 29(2): 671-691 doi: 10.3934/dcds.2011.29.671 +[Abstract](1653) +[PDF](259.9KB)
Abstract:
The paper studies the notion of subdifferentials of functions defined on a time scale. The subdifferential of a given function $f$ is defined as the set of certain extended functions. Since the convexity of the given function guarantees its subdifferentiability, properties of convex functions on time scales are presented. We show that the convexity of a function is the necessary and sufficient condition for its subdifferentiability. The relations between the delta, nabla, diamond-$\alpha$ derivatives and subdifferentials of convex functions are given.

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