The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Stochastic differential equations and diffusion processes

Distributed consensus tracking is addressed in this paper for multi-agent systems with Lipschitz-type node dynamics. The main contribution of this work is solving the consensus tracking problem without the assumption that the topology among followers is strongly connected and fixed. By using tools from M-matrix theory, a class of consensus tracking protocols based only on the relative states among neighboring agents is designed. By appropriately constructing Lyapunov function, it is proved that consensus tracking in the closed-loop multi-agent systems with a fixed topology having a directed spanning tree can be achieved if the feedback gain matrix and the coupling strength are suitably selected. Furthermore, with the assumption that each possible topology contains a directed spanning tree, it is theoretically shown that consensus tracking under switching directed topologies can be achieved if the control parameters are suitably selected and the dwell time is larger than a positive threshold. The results are then extended to the case where the communication topology contains a directed spanning tree only frequently as the system evolves with time. Finally, some numerical simulations are given to verify the theoretical analysis.

Consensus Tracking of Multi-Agent Systems With Lipschitz-Type Node Dynamics and Switching Topologies

Second-order leader-following consensus of nonlinear multi-agent systems via pinning control

Calcul des Probabilités. By Paul Lévy. Pp. 350. Fr. 40. 1925. (Gauthier-Villars.)

A recent method for solving differential equations using feedforward neural networks was applied to a non-steady fixed bed non-catalytic solid–gas reactor As neural networks have universal approximation capabilities, it is possible to postulate them as solutions for a given DE problem that defines an unsupervised error The training was performed using genetic algorithms and the gradient descent method The solution was found with uniform accuracy (MSE ∼10−9) and the trained neural network provides a compact expression for the analytical solution over the entire finite domain The problem was also solved with a traditional numerical method In this case, solution is known only over a discrete grid of points and its computational complexity grows rapidly with the size of the grid Although solutions in both cases are identical, the neural networks approach to the DE problem is qualitatively better since, once the network is trained, it allows instantaneous evaluation of solution at any desired number of points spending negligible computing time and memory

Solving differential equations with unsupervised neural networks

We establish conditions for existence and for nonexistence of nontrivial solutions to an elliptic system of partial dierential equations. This system is of gradient type and has a nonlinearity with critical growth.

Existence of Solutions for Elliptic Systems with Critical Sobolev Exponent

This work is devoted to study the existence of solutions to equations of p-Laplacian type. We prove the existence of at least one solution, and under further assumptions, the existence of infinitely many solutions. In order to apply mountain pass results, we introduce a notion of uniformly convex functional that generalizes the notion of uniformly convex norm.

Mountain pass solutions to equations of p-Laplacian type

A Black-Scholes option pricing model with transaction costs

The existence of weak solutions locally in time to the quantum hydrodynamic equations in bounded domains is shown. These Madelung-type equations consist of the Euler equations, including the quantum Bohm potential term, for the particle density and the particle current density and are coupled to the Poisson equation for the electrostatic potential. This model has been used in the modeling of quantum semiconductors and superfluids. The proof of the existence result is based on a formulation of the problem as a nonlinear Schrodinger–Poisson system and uses semigroup theory and fixed-point techniques.

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Maria C. Mariani

Papers

Solving differential equations with unsupervised neural networks

Existence of Solutions for Elliptic Systems with Critical Sobolev Exponent

Mountain pass solutions to equations of p-Laplacian type

A Black-Scholes option pricing model with transaction costs

Local existence of solutions to the transient quantum hydrodynamic equations