Liouville Integrability in a Four-Dimensional Model of the Visual Cortex

We consider a natural extension of the Petitot–Citti–Sarti model of the primary visual cortex. In the extended model, the curvature of contours is taken into account. The occluded contours are completed via sub-Riemannian geodesics in the four-dimensional space M of positions, orientations, and curvatures. Here, M=R2×SO(2)×R models the configuration space of neurons of the visual cortex. We study the problem of sub-Riemannian geodesics on M via methods of geometric control theory. We prove complete controllability of the system and the existence of optimal controls. By application of the Pontryagin maximum principle, we derive a Hamiltonian system that describes the geodesics. We obtain the explicit parametrization of abnormal extremals. In the normal case, we provide three functionally independent first integrals. Numerical simulations indicate the existence of one more first integral that results in Liouville integrability of the system.

APPLICATION OF THE PONTRYAGIN MAXIMUM PRINCIPLE TO CONTROL TUMOR ANGIOGENESIS

We present a sample application covering several cases using an extension of the Pontryagin Minimum Principle (PMP) [3]. We are interested in the management of tumor angiogenesis, that is, the therapeutic management of the proliferation of cancer cells that develop new blood vessels. Let us formulate the problem and derive the optimal control and apply the Pontryagin maximum principle to our optimal trajectory, and we derive the theorem and check it with an example. Then we will study stabilization.

Optimal Control in a Mathematical Model of Smoking

This paper presents a dynamic model of smoking with optimal control. The mathematical model is divided into 5 sub-classes, namely, non-smokers, occasional smokers, active smokers, individuals who have temporarily stopped smoking, and individuals who have stopped smoking permanently. Four optimal controls, i.e., anti-smoking education campaign, anti-smoking gum, anti-nicotine drug, and government prohibition of smoking in public spaces are considered in the model. The existence of the controls is also presented. The Pontryagin maximum principle (PMP) was used to solve the optimal control problem. The fourth-order Runge-Kutta was employed to gain the numerical solutions.

Necessary optimality condition for the minimal time crisis relaxing transverse condition via regularization

We derive necessary optimality conditions for the time of crisis problem under a more general hypothesis than the usual one encountered in the hybrid setting, which requires that any optimal solution should cross the boundary of the constraint set transversely. Doing so, we apply the Pontryagin Maximum Principle to a sequence of regular optimal control problems whose integral cost approximates the time of crisis. Optimality conditions are derived by passing to the limit in the Hamiltonian system (without the use of the hybrid maximum principle). This convergence result essentially relies on the boundedness of the sequence of adjoint vectors in L∞. Our main contribution is to relate this property to the boundedness in L1 of a suitable sequence which allows to avoid the use of the transverse hypothesis on optimal paths. An example with non-transverse trajectories for which necessary conditions are derived highlights the use of this new condition.

Optimization of aerospace aircraft guidance to the landing area

The article considers a technique for constructing an optimal guidance procedure for an aerospace aircraft. The technique is based on the adaptation of the Pontryagin maximum principle for the considered class of problems. At the same time the guidance accuracy is ensured by solving a boundary value problem, which is periodically performed during the flight. The developed procedure for predicting the final parameters of the optimal flight according to a simplified motion model is presented, which also makes it possible to determine the value of the actual miss. A detailed mathematical description of the proposed technique is given. The feasibility of the proposed technique is ensured by minimizing the amount of computational operations. The guidance algorithm efficiency is illustrated by a numerical example with a flight simulation procedure taking into account all significant factors. The paper also provides examples of solving boundary value problems and the results of modeling the optimal guidance.

Landing Trajectory Design for UAV Considering Control Restrictions and Landing Speed

The article presents a method for designing the trajectory of the UAV in space, taking into account the restriction on control. The chosen optimal controls are namely normal overload with restrictions, tangential overload with restrictions, and lateral overload. The Pontryagin maximum principle allows the transition of the optimal control problem to a boundary value problem. The parameter continuation method is applied to solve the boundary problem. The article results reveal reference trajectories in different cases of UAV landing. This result allows the design of reference trajectories for the UAV to attain the highest landing efficiency.

First order necessary optimal conditions in Gursat-Darboux stochastic systems

For optimal control problems, described by the Gursat-Darboux stochastic system, a number of first-order necessary optimality conditions are formulated and proved, which are the stochastic analogue - the Pontryagin maximum principle, the linearized maximum principle and the Euler equation.

Regularized classical optimality conditions in iterative form for convex optimization problems for distributed Volterra-type systems

We consider the regularization of the classical optimality conditions (COCs) — the Lagrange principle and the Pontryagin maximum principle — in a convex optimal control problem with functional constraints of equality and inequality type. The system to be controlled is given by a general linear functional-operator equation of the second kind in the space $L^m_2$, the main operator of the right-hand side of the equation is assumed to be quasinilpotent. The objective functional of the problem is strongly convex. Obtaining regularized COCs in iterative form is based on the use of the iterative dual regularization method. The main purpose of the regularized Lagrange principle and the Pontryagin maximum principle obtained in the work in iterative form is stable generation of minimizing approximate solutions in the sense of J. Warga. Regularized COCs in iterative form are formulated as existence theorems in the original problem of minimizing approximate solutions. They “overcome” the ill-posedness properties of the COCs and are regularizing algorithms for solving optimization problems. As an illustrative example, we consider an optimal control problem associated with a hyperbolic system of first-order differential equations.

Application of optimal control to tuberculosis model with parameter estimations: Bayesian approach

In this paper, one-strain tuberculosis (TB) model with two control mechanisms, education campaigns and chemoprophylaxis of TB-infected patients, was studied to determine their effects on the reduction of latent and active TB cases. In the case of analysis, boundedness and positivity of the model solutions were carried out to determine the biological feasibility of the study. Besides, the calibration of the parameters by utilizing the identifiability technique through the Markov chain Monte Carlo (MCMC) was thoroughly analysed. The optimum conditions for controlling TB were derived from the Pontryagin Maximum Principle. The numerical simulations were carried out using the forward-back sweep method with the help of the Runge-Kutta fourth-order numerical schemes. Simulation results showed that the education campaigns strategy is more effective in reducing TB infections than the chemoprophylaxis of TB-infected individuals. The combination of the two control strategies reduces a significant number of infections than when each strategy is used on its own. To minimize the transmission of TB from the community, we recommend the education campaigns strategy be a focal point and treatment of latent TB to be paired with the treatment of active TB cases. Keywords: Tuberculosis, Education campaigns, Chemoprophylaxis, MCMC