# Constructive Methods for Initialization and Handling Mixed State-Input Constraints in Optimal Control

10 08 Category : Optimal Control | All

**Authors**: Knut Graichen and Nicolas Petit

AIAA Journal of Guidance, Control, and Dynamics Vol. 31, No. 5, pp 1334-1343. 2008. DOI: 10.2514/1.33870

New methods are presented to address two issues in indirect optimal control: the calculation of a starting point for the numerical solution and the consideration of mixed state-input constraints. In the ﬁrst method, an auxiliary optimal control problem is constructed from a given initial trajectory of the system. Its adjoint variables are simply zero. This auxiliary problem is then used within a homotopy approach to eventually reach the original optimal control problem and the desired optimal solution. The second method concerns the incorporation of mixed state-input constraints into the dynamics of the considered optimal control problem. It uses saturation functions which strictly satisfy the constraints. In this way, the original constrained optimal control problem is transformed into an unconstrained one with an additional regularization term. The two approaches are derived within a general framework. For sake of illustration, they are applied to the space shuttle reentry problem, which represents a challenging benchmark due to its high numerical sensitivity and the presence of input and heating constraints. The reentry problem is solved with a collocation method and demonstrates the applicability and accuracy of the proposed constructive methods.

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**BibTeX**:

@Article{,

author = {K. Graichen, N. Petit},

title = {Constructive Methods for Initialization and Handling Mixed State-Input Constraints in Optimal Control},

journal = {AIAA Journal of Guidance, Control, and Dynamics},

volume = {31},

number = {5},

pages = {1334-1343},

year = {2008},

abstract = {New methods are presented to address two issues in indirect optimal control: the calculation of a starting point for the numerical solution and the consideration of mixed state-input constraints. In the first method, an auxiliary optimal control problem is constructed from a given initial trajectory of the system. Its adjoint variables are simply zero. This auxiliary problem is then used within a homotopy approach to eventually reach the original optimal control problem and the desired optimal solution. The second method concerns the incorporation of mixed state- input constraints into the dynamics of the considered optimal control problem. It uses saturation functions which strictly satisfy the constraints. In this way, the original constrained optimal control problem is transformed into an unconstrained one with an additional regularization term. The two approaches are derived within a general framework. For sake of illustration, they are applied to the space shuttle reentry problem, which represents a challenging benchmark due to its high numerical sensitivity and the presence of input and heating constraints. The reentry problem is solved with a collocation method and demonstrates the applicability and accuracy of the proposed constructive methods.},

location = {},

keywords = {TRAJECTORY OPTIMIZATION; NONLINEAR-SYSTEMS; SPACE-SHUTTLE; ALGORITHM; DESIGN}}