Spatial cognitive function is crucial for the animal’s survival. However, formation of place codes in different dimensional spaces cannot be uniformly explained. In this paper, a constrained optimization model based on information theory constructed to explain cell activity across species. The question proposed as, using limited amount neural energy, how design field obtain most efficient spati...

This paper considers a controlled Itô-Lévy process the information available to the controller is possibly less than the overall information. All the system coefficients and the objective performance functional are allowed to be random, possibly nonMarkovian. Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explic...

We present two versions of the maximum principle for nonsmooth hybrid optimal control problems, the first one of which requires differentiability along the reference trajectory and yields an adjoint equation of the usual kind, while the second one only requires approximability to first order by Lipschitz maps, and yields an adjoint differential inclusion involving a generalized gradient of the ...

In this paper, an optimal singular stochastic control problem is considered. For this model, it is obtained a general stochastic maximum principle by using a time transformation. This is the first version of the stochastic maximum principle that covers the singular control problem in the nonlinear case.

x 1. Introduction Optimal control was born in 1696 |300 years ago this year| in the Netherlands, when Johann Bernoulli challenged his contemporaries with the \brachystochrone problem" (BP). The purpose of this paper is to give a brief outline of why this event truly deserves to be called the birth of optimal control, and how the research that began in 1696 has led to modern optimal control theo...

The continuous dynamical system approach to deep learning is explored in order to devise alternative frameworks for training algorithms. Training is recast as a control problem and this allows us to formulate necessary optimality conditions in continuous time using the Pontryagin’s maximum principle (PMP). A modification of the method of successive approximations is then used to solve the PMP, ...

In his proof of Andreev’s theorem, Thurston in [1] introduced a conformal geometric structure on two dimensional simplicial complexes which is an analogue of a Riemannian metric. He then used a version of curvature to prove the existence of circle-packings (see also Marden-Rodin [2] for more exposition). Techniques very similar to elliptic partial differential equation techniques were used by Y...

subject to (tf , x(tf )) ∈ S = [t0,∞)× S1 where S1 is a k dimensional manifold in Rn S1 = {x ∈ R : h1(x) = h2(x) = · · · = hn−k(x) = 0} where hi are C1 functions from Rn to R subject to ẋ = f(x, u), x(t0) = x0 for u ∈ C[t0, T ] and u(t) ∈ U ⊂ Rm with f and L being C1 functions. Let u∗ : [t0, tf ] → R be an optimal control with state trajectory x∗ : [t0, tf ] → Rn and a constant. Then there exis...

In many practical situations, the Maximum Entropy (MaxEnt) approach leads to reasonable distributions. However, in an important case when all we know is that the value of a random variable is somewhere within the interval, this approach leads to a uniform distribution on this interval – while our intuition says that we should have a distribution whose probability density tends to 0 when we appr...