optimal switching for partial differential equations i abstract optimal switching for an abstract cauchy problem is considered a type of approximating problems with value functions u 0 is studied we prove that each u is the unique viscosity solution of corresponding hamilton jacobi system that u converges to the value function u of the original problem as goes to and that via u an optimal control for the switching problem is determined

w symmetries of ito stochastic differential equations we discuss w symmetries of ito stochastic differential equations introduced in a recent paper by gaeta and spadaro j math phys 58 053503 2017 in particular we discuss the general form of acceptable generators for continuous lie point w symmetry arguing that they are related to the linear conformal group and how w symmetries can be used in the integration of ito stochastic equations along kozlov theory for standard deterministic or random symmetries it turns out that this requires in general considering more general classes of stochastic equations than just ito ones we discuss w symmetries of ito stochastic differential equations introduced in a recent paper by gaeta and spadaro j math phys 58 053503 2017 in particular we discuss the general form of acceptable generators for continuous lie point w symmetry arguing that they are related to the linear conformal group and how w symmetries can be used in the integration of ito stochastic equations along kozlov theory for standard deterministic or random symmetries it turns out that this requires in general considering more general classes of stochastic equations than just ito ones

a convex penalty for switching control of partial differential equations abstract a convex penalty for promoting switching controls for partial differential equations is introduced such controls consist of an arbitrary number of components of which at most one should be simultaneously active using a moreau yosida approximation a family of approximating problems is obtained that is amenable to solution by a semismooth newton method the efficiency of this approach and the structure of the obtained controls are demonstrated by numerical examples

stochastic calculus of variations abstract a theory of stochastic calculus of variations is presented which generalizes the ordinary calculus of variations to stochastic processes generalizations of the euler equation and noether s theorem are obtained and several conservation laws are discussed an application to nelson s probabilistic framework of quantum mechanics is also given

a connection between quantum dynamics and approximation of markov diffusions it is observed that the existence of an attracting set in the class of solutions to the stochastic lagrangian variational principle leads to a natural problem of convergence of diffusions in the carlen class it is then shown how dynamical properties enable one to prove some convergence results in the two dimensional gaussian case

malliavin calculus and euclidean quantum mechanics i functional calculus abstract we give a rigorous version of the functional calculus developed by r feynman in relation to his path integral formulation of quantum mechanics our approach is euclidean but distinct from the one founded on the feynman kac formula it uses two basic ingredients a new probabilistic interpretation of the classical heat equation introduced recently in the framework of euclidean quantum mechanics and an infinite dimensional differential calculus adapted to functionals of the diffusion processes relevant for this interpretation

optimal switching for ordinary differential equations we consider the problem of controlling an ordinary differential equation subject to positive switching costs and show in particular that the value functions form the viscosity solution cf 6 7 of the dynamic programming quasi variational inequalities this interpretation allows for a rigorous application of various dynamic programming techniques

differential games with switching strategies abstract a zero sum differential game of infinite horizon is considered positive switching costs are associated with each player we prove that under a condition which is different from the isaacs condition the elliot kalton value of the game always exists the value is the unique viscosity solution of corresponding isaacs type system of equations which appears to be a system of quasi variational inequalities with bilateral obstacles the discussion of the case that the switching costs approach to zero shows that the isaacs condition still has to be assumed to guarantee that the limit of the values corresponding to positive switching costs converge to the value corresponding to zero switching costs the classical elliot kalton value of the game

towards a quantum analog of weak kam theory we discuss a quantum analogue of mather s minimization principle for lagrangian dynamics and provide some formal calculations suggesting the corresponding euler lagrange equation we then rigorously construct from the dual eigenfunctions of a certain non selfadjoint operator a candidate for a minimizer and recover aspects of weak kam theory in the limit as h 0 regarding our state as a quasimode we furthermore derive some error estimates although it remains an open problem to improve these bounds

hamilton s principle in stochastic mechanics in this paper we establish three variational principles that provide new foundations for nelson s stochastic mechanics in the case of nonrelativistic particles without spin the resulting variational picture is much richer and of a different nature with respect to the one previously considered in the literature we first develop two stochastic variational principles whose hamilton jacobi like equations are precisely the two coupled partial differential equations that are obtained from the schrodinger equation madelung equations the two problems are zero sum noncooperative stochastic differential games that are familiar in the control theory literature they are solved here by means of a new absolutely elementary method based on lagrange functionals for both games the saddle point equilibrium solution is given by the nelson s process and the optimal controls for the two competing players are precisely nelson s current velocity v and osmotic velocity u respectively the first variational principle includes as special cases both the guerra morato variational principle phys rev d 27 1774 1983 and schrodinger original variational derivation of the time independent equation it also reduces to the classical least action principle when the intensity of the underlying noise tends to zero it appears as a saddle point action principle in the second variational principle the action is simply the difference between the initial and final configurational entropy it is therefore a saddle point entropy production principle from the variational principles it follows in particular that both v x t and u x t are gradients of appropriate principal functions in the variational principles the role of the background noise has the intuitive meaning of attempting to contrast the more classical mechanical features of the system by trying to maximize the action in the first principle and by trying to increase the entropy in the second combining the two variational principles we get the quantum hamilton principle i e a variational characterization of the logarithm of the wave function the lagrangian is the lagrangian of classical mechanics with the complex valued velocity v iu replacing the classical velocity the dynamics is given by a stochastic differential equation for real valued diffusions with complex valued drift and driving noise processes from the variational principle we derive a newton type law we finally define the momentum process and show that its mean and variance coincide with those of the quantum momentum operator

geometrical stochastic control and quantization a class of diffusion processes controlled by the geometry of the manifold on which they evolve is considered the kinetic energy of such diffusions is shown to be a geometric object expressed in terms of the curvature and torsion tensors this gives rise to an action functional leading to a variational principle from which a nontrivial critical geometry with nonvanishing torsion emerges the resulting criticality condition is related to the schrodinger equation in a manner that reproduces the features of nelson s stochastic approach to quantum mechanics

on vector quasivariational inequalities in this paper we consider the vector quasivariational inequalities and the vector quasivariational like inequalities for multifunctions with vector values and prove some existence theorems of solutions for our inequalities also we give the relationship between a kind of the vector variational inequality for multifunctions and a vector optimization problem involving nondifferentiable lipschitz functions

hamilton jacobi equations in infinite dimensions i uniqueness of viscosity solutions soient un espace de banach v son dual v et des solutions d une equation h x u du 0 dans etablies dans un sous ensemble de v h v r v r est continue on demontre que des solutions classiques sont determinees de facon unique par leurs valeurs limites on montre que l extension de la notion de solutions viscosite aux espaces de banach reussit dans des espaces munis de la propriete de radon nikodym

lagrangians of stochastic mechanics the stochastic variational principle which applies to diffusions and extremises the mean value of the classical action is generalised to include non markovian diffusions a whole family of different stochastic actions labelled by a time dependent parameter is obtained and all of them are shown to lead to the shrodinger equation as a result the well known degeneracy of the class of stochastic processes which correspond to a single quantum state is recovered it is also related to an arbitrariness in the choice of the noise which can occur in two places in the definition of the noise through the time dependent parameter and in the auto correlations of the noise which although constant during the variational method remain undetermined the whole scheme is given a covariant form

a variational principle for the navier stokes equation abstract motivated from arnold s variational characterization of the euler equation in terms of geodesic families of diffeomorphisms a variational principle for the motion of incompressible viscous fluids is presented a volume preserving diffusion process with drift velocity field subject to the navier stokes equation is shown to extremize the energy functional of the fluid under a certain class of stochastic variations

quantum mechanics and stochastic control theory a time symmetric stochastic control theory is proposed as one of the representatives of quantum mechanics the main idea is based on nelson s probability theoretical approach to quantum mechanics his approach is reformulated as a time symmetric stochastic control problem several different control constraints equivalent to nelson s are obtained one of them has a close connection to the lagrangian formalism of classical mechanics this suggests to us the use of stochastic calculus of variations within the realm of this time symmetric stochastic control theory it is shown why schrodinger s original variational method of quantization was successful several advantageous points of the stochastic control theoretical approach to quantum mechanics including the analysis of the classical limit are also discussed

nonconvex penalization of switching control of partial differential equations abstract this paper is concerned with optimal control problems for parabolic partial differential equations with pointwise in time switching constraints on the control a standard approach to treat constraints in nonlinear optimization is penalization in particular using l 1 type norms applying this approach to the switching constraint leads to a nonsmooth and nonconvex infinite dimensional minimization problem which is challenging both analytically and numerically adding h 1 regularization or restricting to a finite dimensional control space allows showing existence of optimal controls first order necessary optimality conditions are then derived using tools of nonsmooth analysis their solution can be computed using a combination of moreau yosida regularization and a semismooth newton method numerical examples illustrate the properties of this approach

hamilton jacobi equations in infinite dimensions iii on etudie des equations de la forme f x u du 0 dans ou est un sous ensemble ouvert d un espace de banach v la fonction inconnue u r est continue du x est la derivee de frechet de u en x et f est une fonction non lineaire qui est une application continue de r v dans r on introduit la notion de solution viscosite stricte afin d eliminer l hypothese que v a la propriete de radon nikodym

noether s theorem in symmetric stochastic calculus of variations quite recently a symmetric stochastic calculus of variations was proposed to formulate canonical stochastic dynamics which is an extension of nelson s stochastic mechanics in this article a noether s theorem is formulated within this calculus of variations conservation laws of momentum angular momentum and energy are proved which are related with the same laws in quantum mechanics

geometry of the schrodinger equation and stochastic mass transportation the schrodinger equation is deduced from a geometric principle lagrange hamilton jacobi and hamilton formalisms are defined almost analogously to the deterministic case which can be identified as geometric optic the form of these formalisms is identical to the deterministic formalisms furthermore it will be shown how the deterministic case is superficial to the stochastic one if planck s constant is very small an elementary proof of heisenberg s uncertainty relation finishes the paper

optimal switching for partial differential equations ii abstract optimal switching problem for systems governed by abstract semilinear evolution equations is considered the generator of semigroups is allowed to depend on the decision parameter and the nonlinear term in the evolution equation and the integral in the index are allowed to be unbounded under some mild conditions we prove that the value function of the optimal switching problem is the limit of the value functions of approximation problems we also discuss the finite horizontal case and the problem of approximating continuous controls by switching

hamilton jacobi equations in infinite dimensions ii existence of viscosity solutions on demontre des resultats d existence generaux pour 2 problemes 1 le probleme de cauchy u t h x t u du 0 dans v 0 t u x 0 x dans v et 2 le probleme stationnaire u h x u du 0 dans v ou h et sont donnees

on a certain class of two sided continuous local semimartingales toward a sample wise characterization of the nelson process working with the extended framework of stochastic integrals recently discovered by ito a complex of stochastic processes inherent in quantum mechanics the nelson process is characterized in terms of sample paths it is shown that the nelson process belongs to a certain class of two sided continuous local semimartingales several basics of stochastic calculus in this class are presented stochastic calculus of variations is applied in this class to construct the nelson process and to further illustrate some details of its sample paths examples are the bound states the two slit interference and the gravity in quantum mechanics

a numerical method for a class of mixed switching and impulsive optimal control problems in this paper we consider a class of optimal control problems in which the dynamical system involves both switching and impulsive controls the number of switching times are supposed to be finite and without loss of generality to occur together with the impulses we show that this class of optimal control problems is equivalent to a special class of optimal parameter selection problems the gradient formulae for the cost functional and the constraint functional are derived on this basis the control parameterization is shown to be applicable for numerical solution a numerical example is solved for illustration

a system of nonlinear partial differential equations arising in the optimal control of stochastic systems with switching costs the problem of optimal switching control of a diffusion process with costly switchings leads to a system of fully nonlinear elliptic partial differential equations with implicit obstacles we obtain results on the existence uniqueness and regularity of the solution of this system

maupertuis principle of least action in stochastic calculus of variations within the framework of stochastic calculus of variations for time symmetric semimartingales x t we consider two different stochastic versions of maupertuis least action principle in lagrangian and hamiltonian terms the general results are applied to classical statistical mechanics where they coincide with those of classical calculus of variations and to nelson s stochastic mechanics an approach to quantum mechanics where a time symmetric semimartingale represents the position of a particle and the dynamics is expressed by a stochastic version of hamilton s principle of least action some historic examples of old quantum theory are discussed

canonical stochastic dynamical systems a canonical stochastic dynamical system for time symmetric semimartingales is formulated by the stochastic least action principle in a new stochastic calculus of variations a certain class of stochastic dynamical systems gives a hamiltonian formalism of nelson s stochastic mechanics in a manner analogous to classical mechanics the notions of a stochastic poisson bracket and canonical transformation are introduced to the stochastic dynamical systems it is shown that the phase factor of the wave function plays the role of a generating function of the canonical transformation

dynamic programming and quantum mechanical motion abstract in this article we show that quantum dynamics is the most natural generalization of classical dynamics from the point of view of optimal control employing the techniques of dynamic programming we derive the schrodinger equation starting from the lagrangian defined in terms of nelson s forward and backward velocities the generalization to the relativistic case is also analyzed and the klein gordon equation is similarly derived

a simplified variational characterization of schrodinger processes the variational problem e 1 2 t0 t a t xt 2 dt t0c t xt dt min where t is the drift process of a diffusion process with unit diffusion coefficient and given initial and final distributions a is a given vector field and c is a given scalar field is considered it is shown that the solution is given by a certain markovian diffusion process which in the case a 0 c 0 first was investigated by schrodinger sitzungsber preuss akad wiss phys math kl 1931 144

stochastic embedding of dynamical systems most physical systems are modeled by an ordinary or a partial differential equation like the n body problem in celestial mechanics in some cases for example when studying the long term behavior of the solar system or for complex systems there exist elements which can influence the dynamics of the system which are not well modeled or even known one way to take these problems into account consists of looking at the dynamics of the system on a larger class of objects that are eventually stochastic in this paper we develop a theory for the stochastic embedding of ordinary differential equations we apply this method to lagrangian systems in this particular case we extend many results of classical mechanics namely the least action principle the euler lagrange equations and noether s theorem we also obtain a hamiltonian formulation for our stochastic lagrangian systems many applications are discussed at the end of the paper

extremal flows in wasserstein space we develop an intrinsic geometric approach to the calculus of variations in the wasserstein space we show that the flows associated with the schrodinger bridge with general prior with optimal mass transport and with the madelung fluid can all be characterized as annihilating the first variation of a suitable action we then discuss the implications of this unified framework for stochastic mechanics it entails in particular a sort of fluid dynamic reconciliation between bohm s and nelson s stochastic mechanics we develop an intrinsic geometric approach to the calculus of variations in the wasserstein space we show that the flows associated with the schrodinger bridge with general prior with optimal mass transport and with the madelung fluid can all be characterized as annihilating the first variation of a suitable action we then discuss the implications of this unified framework for stochastic mechanics it entails in particular a sort of fluid dynamic reconciliation between bohm s and nelson s stochastic mechanics