## Stationary condition

H(x*, u* , A*, t) < H(x*, u* + Su, A*, t) [55] Boundary condition

For problems with constraints in the control variables like those given by eqn [46], the stationary condition must be modified to include Pontryagin's maximum principle which establishes that the solution values for the constrained control variables must lie along an optimal path. That is, any variation in the optimal control profile u*(t) at time t, while keeping the state and co-state variables z(t), A(t) and M(t) at their optimal values, will force an increase in the value of the Hamiltonian. This replaces the unconstrained minimum condition of eqn [52] and is stated mathematically in eqn [55]. Also, the second term of the boundary condition in eqn [56] vanishes for fixed-time problems.

Solution of the optimization problem of eqns [40]-[46] in its variational formulation requires integration of two sets of differential equations given by [50] and [51] to get the state variables z and adjoint variables A for the ordinary differential equation (ODE). Since these equations are also a function of the control profile u(t), their integration is first performed with guessed values of this vector. Eqns [53] and [54] are used to find the second set of adjoint variables (M) and the slack variables (s2) associated with the constraint on g(u(t), z(t)). Finally, eqn [52] or [55] provides the updated values for u(t), the control profile, whereas the adjoint variables for the boundary conditions are calculated from eqn [56]. The whole procedure involves successive iterations of the control vector and can be computationally intensive, especially for problems with many constraints.

An alternative solution can be formulated to overcome the difficulty posed by the differential constraints. Eqns [40]-[46] are discretized using finite elements. Within each element, function approximation is expressed in terms of orthogonal polynomials and the resulting problem is amenable to a mathematical treatment intended to minimization problems involving only algebraic equations.

Discretization of the optimal control problem leads to the nonlinear problem model given below. This formulation consists of the discretized objective function of the original problem, the continuity equations for state variables and inequality constraints in the b b original formulation.

## Solar Panel Basics

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