example_stability.m

f = @(x) [-x(2); x(1)]; % Example system dynamics
V = @(x) x(1)^2 + x(2)^2; % Example Lyapunov function
h = @(x) 2*norm(x)^2; % Example h(x)

% Define the domain for checking the conditions
domain = [-2:0.1:2; -2:0.1:2]; % Example domain as a 2D grid

% Evaluate stability
isStable = evaluateStability(f, V, h, domain);

% MATLAB Sample Code for Approximating Compositional Lyapunov Functions Using Deep Neural Networks

%% Define Neural Network Architecture for Each Subsystem
% Assuming two subsystems as an example
inputSize = 3; % Example input size (state dimension of the subsystem)
hiddenLayers = [20, 15]; % Example hidden layer sizes
outputSize = 1; % Scalar output representing Lyapunov function value

% Define neural networks for each subsystem
net1 = feedforwardnet(hiddenLayers);
net1 = configure(net1, zeros(inputSize,1), zeros(outputSize,1));

net2 = feedforwardnet(hiddenLayers);
net2 = configure(net2, zeros(inputSize,1), zeros(outputSize,1));

%% Generate Training Data
% Simulate subsystems to generate training data (states and corresponding Lyapunov function values)
% Placeholder for actual simulation data
dataSubsystem1 = rand(100, inputSize); % 100 random samples for subsystem 1
dataSubsystem2 = rand(100, inputSize); % 100 random samples for subsystem 2

% Corresponding Lyapunov function values (placeholder for actual computation)
lyapunovValues1 = rand(100, outputSize); % Example Lyapunov values for subsystem 1
lyapunovValues2 = rand(100, outputSize); % Example Lyapunov values for subsystem 2

%% Training Neural Networks
% Train neural networks using the generated data
[net1, tr1] = train(net1, dataSubsystem1', lyapunovValues1');
[net2, tr2] = train(net2, dataSubsystem2', lyapunovValues2');

%% Compose Global Lyapunov Function
% Example composition of global Lyapunov function from individual networks
% This composition depends on the specific system and needs to be adapted
composeLyapunovFunction = @(x) net1(x(1:inputSize)) + net2(x(inputSize+1:end));

%% Stability Analysis Using Global Lyapunov Function
% Analyze the stability of the overall system using the composed Lyapunov function
% Placeholder for actual system state and stability analysis
systemState = rand(inputSize*2, 1); % Example system state
globalLyapunovValue = composeLyapunovFunction(systemState);

% Stability check - placeholder for actual criteria
if globalLyapunovValue < 1
    disp('System is likely stable under current state.');
else
    disp('System stability is uncertain under current state.');
end