Smoothing approach for a class of nonsmooth optimal control problems

Optimal control approach for discontinuous dynamical systems

“…Direct methods are based on the discretization and parameterization, which can lead to nonlinear programming (NLP) problems. Methods such as pseudo-spectral method, measure theoretical approaches, linearization methods, control parameterization methods, finite difference methods, and time-scaling transformation methods belong to this class (see previous studies [37][38][39][40][41][42][43][44][45][46][47] ). However, indirect methods are based on the Pontryagin minimum principle and Hamiltonian-Jacobi-Bellman equations, which can lead to a problem with initial and boundary conditions (see previous studies [48][49][50] ).…”

Fractional Chebyshev pseudospectral method for fractional optimal control problems

“…This method, in comparison to other methods, has higher accuracy and better results. Up to now, the pseudospectral methods have been utilized by many researchers to solve different types of continuous-time problems (see other works 9,[24][25][26][27][28][29]. In this paper, we consider the shifted Chebyshev-Gauss-Lobatto points as interpolating points and apply the fractional Chebyshev pseudospectral (FCP) method based on a new fractional Lagrange polynomial for indirectly solving FOCPs.…”

A direct method based on the Clenshaw-Curtis formula for fractional optimal control problems

“…So lim N →∞I N (x) = I(x * * (•)).Now, from optimality ofx * and x * * (•), we getI(x * * (•)) ≤ I(x * (•)) = lim i→∞ I Ni (x * ) ≤ lim i→∞ I Ni (x) = lim N →∞ I N (x) = I(x * * (•)).Hence, I(x * (•)) = I(x * * (•)). Therefore, x * (•) is an optimal solution of the problem (13)-(14).Notice that Theorem 5.1 shows that, under relatively mild conditions, if the sequence of optimal solutions of the discrete-time problem (24)-(25) is convergent, it must converges to an optimal solution of the continuous-time problem (13)-(14) (or equivalent problem (8)-(10)).…”

“…All of these approaches occur in the two direct and indirect methods. Direct methods are based on "discritize and then optimize", and indirect methods are based on the Pontryagin minimum principle and Hamiltonian equations (For more details see [14] and references therein).…”

“…(15) Now, we apply the CC formula to approximate the integral in problem (13) and (14). We first use the following fractional Lagrange interpolating polynomial…”