The dycon platform is a collection of common tools for the investigation of differential equations, in the context of the development of the Dycon project.
I would change the name Classical, it seems that it has to work with this name, while it does not.HE MIRADO EL LIBRO QUE ME MANDASTE PERO NO HE VISTO ESTO DE LA APROXIMACIÓN. NO VEO PORQUÉ NO HACEMOS BIEN EL ADJUNTO. AHORA MISMO NO TENGO MUCHO MAS TIEMPO PARA MIRAR EN DETALLE EL LIBRO
ClassicalDescent
This method is used within the GradientMethod method. GradientMethod executes iteratively this rutine in order to get one update of the control in each iteration. In the case of choosing ClassicalDescent this function updates the control of the following way:
$$u{old}=u{new}-\alpha dJ$$
where dJ is an approximation of the gradient of J that has been obtained considering the adjoint state of the optimality condition of Pontryagin principle. The optimal control problem is defined by
$$\min J=\min\Psi (t,Y(T))+\int^T_0 L(t,Y,U)dt$$
subject to:
$$\frac{d}{dt}Y=f(t,Y,U).$$
The gradient of $$J$$ is:
$$dJ=\partial_u H=\partial_uL+p\partial_uf$$
An approximation $$p$$ is computed using:
$$-\frac{d}{dt}p = f_Y (t,Y,U)p+L_Y(Y,U)$$
$$ p(T)=\psi_Y(Y(T))$$
Since one the expression of the gradient, we can start with an initial control, solve the adjoint problem and evaluate the gradient. Then we will update the initial control in the direction of the approximate gradient with a step size $$\alpha$$.
In this routine the user has to choose the step size.
WARNING: Using this routine the GradientMethod might not converge if the stepsize is not choosen properly or being slow if the step size is choosen very small. For an adaptative stepsize with Armijo Rule guaranteeing the convergence see (adaptative stepsize).
This routine will tell to GradientMethod to stop when the minimum tolerance of the derivative (or the relative error, user's choice) is reached. Moreover there is a maximum of iterations allowed.
MANDATORY INPUTS:
NAME: iCP
DESCRIPTION: Control problem object, it carries all the information about the dynamics, the functional to be minimized and moreover the updates of the current best control find so far.
CLASS: ControlProblem
NAME: tol
DESCRIPTION: the tolerance desired.
CLASS: double
OPTIONAL INPUT PARAMETERS
NAME: LengthStep
DESCRIPTION: This paramter is the length step of the gradient method that is going to be used. By default, this is 0.1.
CLASS: double
OUTPUT PARAMETERS
Name: Unew
Description: Update of the Control
class: a vector valued function in a form of a double matrix
Name: Ynew
Description: Update of State Vector
class: a vector valued function in a form of a double matrix
Name: Jnew
Description: New Value of functional
class: double
Name: dJnew
Description: New Value of gradient
class: a vector valued function in a form of a double matrix
Name: error
Description: Nthe error |dJ|/|U| or |dJ| depending on the choice of the user.
class: double
Name: stop
Description: if this parameter is true, the routine will tell to GradientMethod to stop
class: logical
I would change the name Classical, it seems that it has to work with this name, while it does not. HE MIRADO EL LIBRO QUE ME MANDASTE PERO NO HE VISTO ESTO DE LA APROXIMACIÓN. NO VEO PORQUÉ NO HACEMOS BIEN EL ADJUNTO. AHORA MISMO NO TENGO MUCHO MAS TIEMPO PARA MIRAR EN DETALLE EL LIBRO
ClassicalDescent
This method is used within the GradientMethod method. GradientMethod executes iteratively this rutine in order to get one update of the control in each iteration. In the case of choosing ClassicalDescent this function updates the control of the following way:
$$u{old}=u{new}-\alpha dJ$$
where dJ is an approximation of the gradient of J that has been obtained considering the adjoint state of the optimality condition of Pontryagin principle. The optimal control problem is defined by $$\min J=\min\Psi (t,Y(T))+\int^T_0 L(t,Y,U)dt$$
subject to:
$$\frac{d}{dt}Y=f(t,Y,U).$$
The gradient of $$J$$ is:
$$dJ=\partial_u H=\partial_uL+p\partial_uf$$
An approximation $$p$$ is computed using:
$$-\frac{d}{dt}p = f_Y (t,Y,U)p+L_Y(Y,U)$$
$$ p(T)=\psi_Y(Y(T))$$
Since one the expression of the gradient, we can start with an initial control, solve the adjoint problem and evaluate the gradient. Then we will update the initial control in the direction of the approximate gradient with a step size $$\alpha$$. In this routine the user has to choose the step size.
WARNING: Using this routine the GradientMethod might not converge if the stepsize is not choosen properly or being slow if the step size is choosen very small. For an adaptative stepsize with Armijo Rule guaranteeing the convergence see (adaptative stepsize).
This routine will tell to GradientMethod to stop when the minimum tolerance of the derivative (or the relative error, user's choice) is reached. Moreover there is a maximum of iterations allowed.
MANDATORY INPUTS:
NAME: iCP DESCRIPTION: Control problem object, it carries all the information about the dynamics, the functional to be minimized and moreover the updates of the current best control find so far. CLASS: ControlProblem
NAME: tol DESCRIPTION: the tolerance desired. CLASS: double
OPTIONAL INPUT PARAMETERS
NAME: LengthStep DESCRIPTION: This paramter is the length step of the gradient method that is going to be used. By default, this is 0.1. CLASS: double
OUTPUT PARAMETERS
Name: Unew Description: Update of the Control class: a vector valued function in a form of a double matrix
Name: Ynew Description: Update of State Vector class: a vector valued function in a form of a double matrix
Name: Jnew Description: New Value of functional class: double
Name: dJnew Description: New Value of gradient class: a vector valued function in a form of a double matrix
Name: error Description: Nthe error |dJ|/|U| or |dJ| depending on the choice of the user. class: double
Name: stop Description: if this parameter is true, the routine will tell to GradientMethod to stop class: logical
Citations:
[1] Cohen, William C.,Optimal control theory—an introduction, Donald E. Kirk, Prentice Hall, Inc., New York (1971), 452 poges. \$13.50,https://onlinelibrary.wiley.com/doi/abs/10.1002/aic.690170452