Hamiltonian Inclusion Research Articles

Non-degenerate necessary conditions for non-convex differential inclusions

This paper concerns dynamic optimization problems with pathwise state constraints, in which the dynamics are formulated as a differential inclusion. First order necessary conditions of optimality are available for these problems. Foremost among these conditions is the Euler Lagrange inclusion (combined with a transversality and Weierstrass conditions), which improves on earlier conditions such as Clarke's Hamiltonian inclusion. The most recent version of the Euler Lagrange inclusion (which we call the standard Euler Lagrange condition) has been validated under unrestrictive conditions, which allow the velocity sets of the differential inclusion to be non-convex. If the initial state is fixed and located in the boundary of the state constraint set then the Euler Lagrange inclusion conveys no information about minimizers. This is the degeneracy phenomenon for state constrained dynamic optimization. The non-degenerate necessary conditions literature provides additional conditions, to supplement the standard necessary conditions. These typically identify some properties of the boundary values of the maximized Hamiltonian which ensure non-triviality of the necessary conditions, under a simple controllability hypothesis. Non-degenerate conditions, incorporating the Euler Lagrange inclusion are currently available only under the hypothesis that the velocity sets are convex. Using new perturbation techniques, we derive a non-degenerate Euler Lagrange condition that is valid for non-convex velocity sets. These conditions bring into line non-degenerate optimality conditions for differential inclusion problems with those for problems involving controlled differential equations which have previously been validated for non-convex velocity sets.

Optimal control of higher order differential inclusions with functional constraints

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

On an Optimal Control Problem with Discontinuous Integrand

We consider an optimal control problem for an autonomous differential inclusion with free terminal time and a mixed functional which contains the characteristic function of a given open set M ⊂ ℝn in the integral term. The statement of the problem weakens the statement of the classical optimal control problem with state constraints to the case where the presence of admissible trajectories of the system in the set M is physically allowed but undesirable due to safety or instability reasons. Using an approximation approach, necessary conditions for the optimality of an admissible trajectory are obtained in the form of Clarke’s Hamiltonian inclusion. The result involves a nonstandard stationarity condition for the Hamiltonian. As in the case of the problem with a state constraint, the obtained necessary optimality conditions may degenerate. Conditions guaranteeing their nondegeneracy and pointwise nontriviality are presented. The results obtained extend the author’s previous results to the case of a problem with free terminal time and more general functional.

Nuclear structure of low-lying states in 60,62,64,66Zn — A shell model description

Shell model calculation has been performed for even–even [Formula: see text]Zn using NuShellX code in [Formula: see text] model space with two different effective Hamiltonians, viz. JUN45 and jj44b. The low-lying structure is studied up to angular momentum, [Formula: see text] = 10[Formula: see text] by calculating level energies, reduced transition probabilities, occupation numbers, lifetimes, and quadrupole moments. The results of the calculations are compared with the available experimental data. It is observed that the inclusion of 1[Formula: see text] orbital in the model space is essential to understand nuclear structure in these isotopes. Shell model calculation with an improved set of effective Hamiltonian parameters and inclusion of [Formula: see text] orbital in the model space are necessary in order to produce finer agreement with the experimental observations.

Regularity of the Hamiltonian Along Optimal Trajectories

This paper concerns state constrained optimal control problems, in which the dynamic constraint takes the form of a differential inclusion. If the differential inclusion does not depend on time, then the Hamiltonian, evaluated along the optimal state trajectory and the co-state trajectory, is independent of time. If the differential inclusion is Lipschitz continuous, then the Hamitonian, evaluated along the optimal state trajectory and the co-state trajectory, is Lipschitz continuous. These two well-known results are examples of the following principle: the Hamiltonian, evaluated along the optimal state trajectory and the co-state trajectory, inherits the regularity properties of the differential inclusion, regarding its time dependence. We show that this principle also applies to another kind of regularity: if the differential inclusion has bounded variation with respect to time, then the Hamiltonian, evaluated along the optimal state trajectory and the co-state trajectory, has bounded variation. Two applications of these newly found properties are demonstrated. One is to derive improved conditions which guarantee the nondegeneracy of necessary conditions of optimality, in the form of a Hamiltonian inclusion. The other application is to derive new, less restrictive, conditions under which minimizers in the calculus of variations have bounded slope. The analysis is based on a new, local, concept of differential inclusions that have bounded variation with respect to the time variable, in which conditions are imposed on the multifunction involved, only in a neighborhood of a given state trajectory.

The Hamiltonian Inclusion for Nonconvex Velocity Sets

Since Clarke's 1973 proof of the Hamiltonian inclusion for optimal control problems with convex velocity sets, there has been speculation (and, more recently, speculation relating to a stronger, partially convexified version of the Hamiltonian inclusion) as to whether these necessary conditions are valid in the absence of the convexity hypothesis. The issue was in part resolved by Clarke himself when, in 2005, he showed that $L^{\infty}$ local minimizers satisfy the Hamiltonian inclusion. In this paper it is shown, by counterexample, that the Hamiltonian inclusion (and so also the stronger partially convexified Hamiltonian inclusion) are not in general valid for nonconvex velocity sets when the local minimizer in question is merely a $W^{1,1}$ local minimizer, not an $L^{\infty}$ local minimizer. The counterexample demonstrates that the need to consider $L^{\infty}$ local minimizers, not $W^{1,1}$ local minimizers, in the proof of the Hamiltonian inclusion for nonconvex velocity sets is fundamental, not just a technical restriction imposed by currently available proof techniques. The paper also establishes the validity of the partially convexified Hamiltonian inclusion for $W^{1,1}$ local minimizers under a normality assumption, thereby correcting earlier assertions in the literature.

Convexity in Zero-Sum Differential Games

A new approach to two-player zero-sum differential games with convex-concave cost function is presented. It employs the tools of convex and variational analysis. A necessary and sufficient condition on controls to be an open-loop saddle point of the game is given. Explicit formulas for saddle controls are derived in terms of the subdifferential of the function conjugate to the cost. Existence of saddle controls is concluded under very general assumptions, not requiring the compactness of control sets. A Hamiltonian inclusion, new to the field of differential games, is shown to describe equilibrium trajectories of the game.

Necessary conditions for optimal control problems with state constraints

Necessary conditions of optimality are derived for optimal control problems with pathwise state constraints, in which the dynamic constraint is modelled as a differential inclusion. The novel feature of the conditions is the unrestrictive nature of the hypotheses under which these conditions are shown to be valid. An Euler Lagrange type condition is obtained for problems where the multifunction associated with the dynamic constraint has values possibly unbounded, nonconvex sets and satisfies a mild ‘one-sided’ Lipschitz continuity hypothesis. We recover as a special case the sharpest available necessary conditions for state constraint free problems proved in a recent paper by Ioffe. For problems where the multifunction is convex valued it is shown that the necessary conditions are still valid when the one-sided Lipschitz hypothesis is replaced by a milder, local hypothesis. A recent ‘dualization’ theorem permits us to infer a strengthened form of the Hamiltonian inclusion from the Euler Lagrange condition. The necessary conditions for state constrained problems with convex valued multifunctions are derived under hypotheses on the dynamics which are significantly weaker than those invoked by Loewen and Rockafellar to achieve related necessary conditions for state constrained problems, and improve on available results in certain respects even when specialized to the state constraint free case. Proofs make use of recent ‘decoupling’ ideas of the authors, which reduce the optimization problem to one to which Pontryagin’s maximum principle is applicable, and a refined penalization technique to deal with the dynamic constraint.

Necessary conditions for functional differential inclusions

We consider an optimal control problem in which the dynamic equation and cost function depend on the recent past of the trajectory. The regularity assumed in the basic data is Lipschitz continuity with respect to the sup norm. It is shown that, for a given optimal solution, an adjoint arc of bounded variation exists that satisfies an associated Hamiltonian inclusion. From this result, known smooth versions of the Pontryagin maximum principle for hereditary problems can be easily derived. Problems with Euclidean endpoint constraints are also considered.

Optimal Control of Unbounded Differential Inclusions

A Mayer problem of optimal control, whose dynamic constraint is given by a convex-valued differential inclusion, is considered. Both state and endpoint constraints are involved. Necessary conditions are proved incorporating the Hamiltonian inclusion, the Euler–Lagrange inclusion, and the Weierstrass–Pontryagin maximum condition. These results weaken the hypotheses and strengthen the conclusions of earlier works. Their main focus is to allow the admissible velocity sets to be unbounded, provided they satisfy a certain continuity hypothesis. They also sharpen the assertion of the Euler–Lagrange inclusion by replacing Clarke’s subgradient of the essential Lagrangian with a subset formed by partial convexification of limiting subgradients. In cases where the velocity sets are compact, the traditional Lipschitz condition implies the continuity hypothesis mentioned above, the assumption of “integrable boundedness” is shown to be superfluous, and this refinement of the Euler–Lagrange inclusion remains a strict improvement on previous forms of this condition.

Pontryagin type conditions for differential inclusions with free time

Optimality conditions for differential inclusion problems, due to Kaskosz and Lojasiewicz, involve a costate equation and a pointwise maximizing property of the optimal velocity, expressed in terms of a Carathéodory selection of the differential inclusion. Such conditions have been extended in various directions, notably to permit unilateral state constraints. Here we add to earlier extensions, principally by allowing free endtimes. This is accomplished even though the data are required to be merely measurable in the time variable. The results are obtained by applying recent optimality conditions for free time problems, involving a Hamiltonian inclusion, to an auxiliary problem and a simple limiting argument.

The adjoint arc in nonsmooth optimization

We extend the theory of necessary conditions for nonsmooth problems of Bolza in three ways: first, we incorporate state constraints of the intrinsic type x ( t ) ∈ X ( t ) x(t) \in X(t) for all t t ; second, we make no assumption of calmness or normality; and third, we show that a single adjoint function of bounded variation simultaneously satisfies the Hamiltonian inclusion, the Euler-Lagrange inclusion, and the Weierstrass-Pontryagin maximum condition, along with the usual transversality relations.

Coextremals and the value function for control problems with data measurable in time

Suppose x ∗(·) is a solution to an optimal control problem formulated in terms of a differential inclusion. Known first-order necessary conditions of optimality assert existence of a coextremal, or adjoint function, p(·), which together with x ∗(·) satisfies the Hamiltonian inclusion and associated transversality condition. In this paper we interpret extremals in terms of generalized gradients of the value function V by demonstrating that p(·) can in addition be chosen to satisfy (p(t) · x ̇ ∗(t), −p(t)) ϵ ∂V(t, x ∗(t)) , a.e. The hypothesis imposed are more or less the weakest under which the Hamiltonian inclusion condition is known to apply and permit, in particular, measurable time dependence of the data. The proof of the results relies on recent developments in Hamilton Jacobi theory applicable in such circumstances. An analogous result is proved for problems where the dynamics are modelled by a differential equation with control term.

Pontryagin-type necessary conditions for differential inclusion problems

We give a new proof of the Pontryagin-type adjoint equation and maximum condition for differential inclusion problems first established by Kaskosz and Lojasiewicz. Our argument is much simpler than the original proof based on generalized control systems, and has the further advantage of illustrating the relationship between the separated necessary conditions and the familiar Hamiltonian inclusion due to Clarke.

Sufficient Optimality Conditions for Stratified Control Problems

Sufficient optimality conditions of dynamic programming type avoiding the axioms of Boltyan-skii’s “regular synthesis” for control problems that have weakly stratified Hamiltonians are proved. An improved version of Boltyanskii’s “fundamental lemma” is applied to the “value function” defined as the minimum of the cost functional along the solutions of a Hamiltonian inclusion which plays the role of a “system of characteristics” for the Hamilton–Jacobi–Bellman equation of dynamic programming.

Optimal Control and the True Hamiltonian

The natural Hamiltonian function in optimal control is generally not differentiable. However, it is possible to use the theory of generalized gradients (which we discuss as a preliminary) to obtain necessary conditions in the form of a “Hamiltonian inclusion”. The advantages of using the true Hamiltonian include the possibility of considering control problems that do not lie within the usual formulation, or problems with nonsmooth data, and the possibility of bringing to bear upon control problems certain Hamiltonian methods from the classical calculus of variations.