The Classical Continuous Mixed Optimal Control of Couple Nonlinear Parabolic Partial Differential Equations with State Constraints

In this work, the classical continuous mixed optimal control vector (CCMOPCV) problem of couple nonlinear partial differential equations of parabolic (CNLPPDEs) type with state constraints (STCO) is studied. The existence and uniqueness theorem (EXUNTh) of the state vector solution (SVES) of the CNLPPDEs for a given CCMCV is demonstrated via the method of Galerkin (MGA). The EXUNTh of the CCMOPCV ruled with the CNLPPDEs is proved. The Frechet derivative (FÉDE) is obtained. Finally, both the necessary and the sufficient theorem conditions for optimality (NOPC and SOPC) of the CCMOPCV with state constraints (STCOs) are proved through using the Kuhn-Tucker-Lagrange (KUTULA) multipliers theorem (KUTULATH).

Fundamentals of Narrowband Array Signal Processing

Array signal processing is an actively developing research area connected to the progress in optimization theory, and remains the key technological development that attracts prevalent attention in signal processing. This chapter provides an overview of the fundamental concepts and essential terminologies employed in narrowband array signal processing. We first develop a general signal model for narrowband adaptive arrays and discuss the beamforming operation. We next introduce the basic performance parameters of adaptive arrays and the second order statistics of the array data. We then formulate various optimal weigh vector solution criteria. Finally, we discuss various types of adaptive filtering algorithms. Besides, this chapter emphasizes the theory of narrowband array signal processing employed in narrowband beamforming and direction-of-arrival (DOA) estimation algorithms.

Boundary Optimal Control for Triple Nonlinear Hyperbolic Boundary Value Problem with State Constraints

The paper is concerned with the state and proof of the solvability theorem of unique state vector solution (SVS) of triple nonlinear hyperbolic boundary value problem (TNLHBVP), via utilizing the Galerkin method (GAM) with the Aubin theorem (AUTH), when the boundary control vector (BCV) is known. Solvability theorem of a boundary optimal control vector (BOCV) with equality and inequality state vector constraints (EINESVC) is proved. We studied the solvability theorem of a unique solution for the adjoint triple boundary value problem (ATHBVP) associated with TNLHBVP. The directional derivation (DRD) of the "Hamiltonian"(DRDH) is deduced. Finally, the necessary theorem (necessary conditions "NCOs") and the sufficient theorem (sufficient conditions" SCOs"), together denoted as NSCOs, for the optimality (OP) of the state constrained problem (SCP) are stated and proved.

A Dynamic Motion Analysis of a Six-Wheel Ground Vehicle for Emergency Intervention Actions

To protect the personnel of the intervention units operating in high-risk areas, it is necessary to introduce (autonomous/semi-autonomous) robotic intervention systems. Previous studies have shown that robotic intervention systems should be as versatile as possible. Here, we focused on the idea of a robotic system composed of two vectors: a carrier vector and an operational vector. The proposed system particularly relates to the carrier vector. A simple analytical model was developed to enable the entire robotic assembly to be autonomous. To validate the analytical-numerical model regarding the kinematics and dynamics of the carrier vector, two of the following applications are presented: intervention for extinguishing a fire and performing measurements for monitoring gamma radiation in a public enclosure. The results show that the chosen carrier vector solution, i.e., the ground vehicle with six-wheel drive, satisfies the requirements related to the mobility of the robotic intervention system. In addition, the conclusions present the elements of the kinematics and dynamics of the robot.

Asymptotic expansion of the ground state energy for nonlinear Schrödinger system with three wave interaction

<p style='text-indent:20px;'>In this paper, we consider the asymptotic behavior of the ground state and its energy for the nonlinear Schrödinger system with three wave interaction on the parameter <inline-formula><tex-math id="M1">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M2">\begin{document}$ \gamma \to \infty $\end{document}</tex-math></inline-formula>. In addition we prove the existence of the positive threshold <inline-formula><tex-math id="M3">\begin{document}$ \gamma^* $\end{document}</tex-math></inline-formula> such that the ground state is a scalar solution for <inline-formula><tex-math id="M4">\begin{document}$ 0 \le \gamma &lt; \gamma^* $\end{document}</tex-math></inline-formula> and is a vector solution for <inline-formula><tex-math id="M5">\begin{document}$ \gamma &gt; \gamma^* $\end{document}</tex-math></inline-formula>.</p>

A Novel Technique to Solve the Fuzzy System of Equations

The aim of this research is to apply a novel technique based on the embedding method to solve the n × n fuzzy system of linear equations (FSLEs). By using this method, the strong fuzzy number solutions of FSLEs can be obtained in two steps. In the first step, if the created n × n crisp linear system has a non-negative solution, the fuzzy linear system will have a fuzzy number vector solution that will be found in the second step by solving another created n × n crisp linear system. Several theorems have been proved to show that the number of operations by the presented method are less than the number of operations by Friedman and Ezzati’s methods. To show the advantages of this scheme, two applicable algorithms and flowcharts are presented and several numerical examples are solved by applying them. Furthermore, some graphs of the obtained results are demonstrated that show the solutions are fuzzy number vectors.

Constraints Optimal Control Governing by Triple Nonlinear Hyperbolic Boundary Value Problem

The focus of this work lies on proving the existence theorem of a unique state vector solution (Stvs) of the triple nonlinear hyperbolic boundary value problem (TNHBVP) when the classical continuous control vector (CCCVE) is fixed by using the Galerkin method (Galm), proving the existence theorem of a unique constraints classical continuous optimal control vector (CCCOCVE) with vector state constraints (equality EQVC and inequality INEQVC). Also, it consists of studying for the existence and uniqueness adjoint vector solution (Advs) of the triple adjoint vector equations (TAEqs) associated with the considered triple state equations (Tsteqs). The Fréchet Derivative (Frde.) of the Hamiltonian (HAM) is found. At the end, the theorems for the necessary conditions and the sufficient conditions of optimality (Necoop and Sucoop) are achieved.

The Continuous Classical Boundary Optimal Control of a Couple Linear Of Parabolic Partial Differential Equations

In this paper the continuous classical boundary optimal problem of a couple linear partial differential equations of parabolic type is studied, The Galerkin method is used to prove the existence and uniqueness theorem of the state vector solution of a couple linear parabolic partial differential equations for given (fixed) continuous classical boundary control vector. The proof of the existence theorem of a continuous classical optimal boundary control vector associated with the couple linear parabolic is given. The Frechet derivative is derived; finally we give a proof of the necessary conditions for optimality (boundary control) of the above problem.