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.

DESAIN KONTROL PENGOBATAN PADA MODEL SIRD UNTUK PENYEBARAN VIRUS COVID-19 MENGGUNAKAN BACKSTEPPING

In this article, we present a control design on a SIRD model with treatment in infected individuals. The SIRD model with treatment is obtained from literature study and the parameter model is obtained from covid-19 daily case in the Riau province using the Particle Swarm Optimization method. The control design is carried out based on the backstepping method combined with feedback linearization based on input and output (IOFL). The SIRD model which is a nonlinear system will be transformed into a normal form using IOFL. Each variable is then stabilized Lyapunov using virtual control which at the same time generates a new state variable. This stage will be carried out iteratively until the last state variable is stabilized using a real control function. This control function is then applied to the SIRD model using the inverse of IOFL transformation. The simulation results compared with the Pontryagin Minimum Principle (PMP) method show that by selecting the appropriate control parameters, backstepping obtains better control performance which is a smaller number of infected populations.

Mathematical Analysis and Optimal Control of Giving up the Smoking Model

In this study, we are going to explore mathematically the dynamics of giving up smoking behavior. For this purpose, we will perform a mathematical analysis of a smoking model and suggest some conditions to control this serious burden on public health. The model under consideration describes the interaction between the potential smokers P , the occasional smokers L , the chain smokers S , the temporarily quit smokers Q T , and the permanently quit smokers Q P . Existence, positivity, and boundedness of the proposed problem solutions are proved. Local stability of the equilibria is established by using Routh–Hurwitz conditions. Moreover, the global stability of the same equilibria is fulfilled through using suitable Lyapunov functionals. In order to study the optimal control of our problem, we will take into account a two controls’ strategy. The first control will represent the government prohibition of smoking in public areas which reduces the contact between nonsmokers and smokers, while the second will symbolize the educational campaigns and the increase of cigarette cost which prevents occasional smokers from becoming chain smokers. The existence of the optimal control pair is discussed, and by using Pontryagin minimum principle, these two optimal controls are characterized. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are performed in order to check the equilibria stability, confirm the theoretical findings, and show the role of optimal strategy in controlling the smoking severity.

Real-Time Velocity Optimization for Energy-Efficient Control of Connected and Automated Vehicles

This paper presents a fast numerical algorithm for velocity optimization based on the Pontryagin' minimum principle (PMP). Considering the difficulties in the application of the PMP when state constraints exist, the penalty function approach is proposed to convert the state-constrained problem into an unconstrained one. Then this paper proposes an iterative numerical algorithm by using the explicit solution to find the optimal solution. The proposed numerical algorithm is applied to the velocity trajectory optimization for energy-efficient control of connected and automated vehicles (CAVs). Simulation results indicate that the algorithm can generate the optimal inputs in milliseconds, and a significant improvement in computational efficiency compared with traditional methods (a few seconds). Hardware in the Loop test for experimental validation is given to further verify the real-time performance of the proposed algorithm.

A minimum-time obstacle-avoidance path planning algorithm for unmanned aerial vehicles

In this article, we present a new strategy to determine an unmanned aerial vehicle trajectory that minimizes its flight time in presence of avoidance areas and obstacles. The method combines classical results from optimal control theory, i.e. the Euler-Lagrange Theorem and the Pontryagin Minimum Principle, with a continuation technique that dynamically adapts the solution curve to the presence of obstacles. We initially consider the two-dimensional path planning problem and then move to the three-dimensional one, and include numerical illustrations for both cases to show the efficiency of our approach.

Optimal Control of a Mathematical Model of Smoking with Temporary Quitters and Permanent Quitters

This article discusses the optimal control of a mathematical model on smoking. This model consists of six population classes, namely potential to become smoker snuffing class irregular smokers regular smokers temporary quitters and permanent quitters The completion of this research uses the Pontryagin minimum principle and numerically using the forward-backward Sweep method. Numerical simulations of the optimal problem show that with the implementation of education campaigns and anti-nicotine medicine, the smokers can be decreased more quickly and the smoking population who quit permanently can be increased. The implementation of both through large amounts needs to be done from the beginning. The use of control in the form of education campaigns is of great value until the end of the research period means that it needs to be done continuously to reduce the number of smokers in the population.

On asymptotically stabilizing control of nonlinear fractional control systems using an optimization scheme

In this paper, stabilization of the nonlinear fractional order systems with unknown control coefficients is considered where the dynamic control system depends on the Caputo fractional derivative. Related to the nonlinear fractional control (NFC) system, an infinite-horizon optimal control (OC) problem is first proposed. It is shown that the obtained OC problem can be an asymptotically stabilizing control for the NFC system. Using the help of an approximation, the Caputo derivative is replaced with the integer order derivative. The achieved infinite-horizon OC problem is then converted into an equivalent finite-horizon one. According to the Pontryagin minimum principle for OC problems and by constructing an error function, an unconstrained minimization problem is defined. In the optimization problem, trial solutions are used for state, costate and control functions where these trial solutions are constructed by using a two-layered perceptron neural network. A learning algorithm with convergence properties is also provided. Two numerical results are introduced to explain the main results.

Optimal Control of Mathematical Models on The Dynamics Spread of Drug Abuse

This article examines the optimal control of a mathematical model of the spread of drug abuse. This model consists of five population classes, namely susceptible to using drugs (S), light-grade drugs (A), heavy-grade drugs (H), medicated drugs (T), and Recovery from drugs (R). The system is solved using the Pontryagin minimum principle and numerically by the forward-backward sweep method. Numerical simulations of the optimal problem show that with the implementation of anti-drug campaigns and strengthening of self-psychology through counseling, the spread of drug abuse can be eradicated more quickly. The implementation of campaigns and strengthening of self-psychology through large amounts of counseling needs to be done from the beginning then the proportion can be reduced until a certain time does not need to be given anymore. The use of control in the form of strengthening efforts to self-psychology through counseling means that it needs to be done in a longer time to prevent the spread of drug abuse.

Pontryagin Neural Networks with Functional Interpolation for Optimal Intercept Problems

In this work, we introduce Pontryagin Neural Networks (PoNNs) and employ them to learn the optimal control actions for unconstrained and constrained optimal intercept problems. PoNNs represent a particular family of Physics-Informed Neural Networks (PINNs) specifically designed for tackling optimal control problems via the Pontryagin Minimum Principle (PMP) application (e.g., indirect method). The PMP provides first-order necessary optimality conditions, which result in a Two-Point Boundary Value Problem (TPBVP). More precisely, PoNNs learn the optimal control actions from the unknown solutions of the arising TPBVP, modeling them with Neural Networks (NNs). The characteristic feature of PoNNs is the use of PINNs combined with a functional interpolation technique, named the Theory of Functional Connections (TFC), which forms the so-called PINN-TFC based frameworks. According to these frameworks, the unknown solutions are modeled via the TFC’s constrained expressions using NNs as free functions. The results show that PoNNs can be successfully applied to learn optimal controls for the class of optimal intercept problems considered in this paper.

An Intrinsic Material Tailoring Approach for Functionally Graded Axisymmetric Hollow Bodies Under Plane Elasticity

One of the main requirements in the design of structures made of functionally graded materials is their best response when used in an actual environment. This optimum behaviour may be achieved by searching for the optimal variation of the mechanical and physical properties along which the material compositionally grades. In the works available in the literature, the solution of such an optimization problem usually is obtained by searching for the values of the so called heterogeneity factors (characterizing the expression of the property variations) such that an objective function is minimized. Results, however, do not necessarily guarantee realistic structures and may give rise to unfeasible volume fractions if mapped into a micromechanical model. This paper is motivated by the confidence that a more intrinsic optimization problem should a priori consist in the search for the constituents’ volume fractions rather than tuning parameters for prefixed classes of property variations. Obtaining a solution for such a class of problem requires tools borrowed from dynamic optimization theory. More precisely, herein the so-called Pontryagin Minimum Principle is used, which leads to unexpected results in terms of the derivative of constituents’ volume fractions, regardless of the involved micromechanical model. In particular, along this line of investigation, the optimization problem for axisymmetric bodies subject to internal pressure and for which plane elasticity holds is formulated and analytically solved. The material is assumed to be functionally graded in the radial direction and the goal is to find the gradation that minimizes the maximum equivalent stress. A numerical example on internally pressurized functionally graded cylinders is also performed. The corresponding solution is found to perform better than volume fraction profiles commonly employed in the literature.