Condición de Legendre–Clebsch bajo hipótesis de rango débil en problemas de control óptimo con restricciones mixtas

Jesús Rodríguez Flores

Resumen


Este trabajo aborda problemas de control con restricciones mixtas, en los que las condiciones clásicas basadas en la condición de independencia lineal de restricciones (LICQ) y en la condición de Legendre–Clebsch sobre todo el espacio de controles pueden resultar excesivamente conservadoras. Se propone un marco de segundo orden basado en una hipótesis de rango débil que permite gradientes activos linealmente dependientes, con el objetivo de vincular la formulación de la segunda variación en programación no lineal con una condición de Legendre–Clebsch formulada en el espacio de controles. A partir de la segunda variación se introduce un cono crítico de direcciones factibles y un subespacio de direcciones críticas del control, y se demuestra que la no negatividad de la forma cuadrática en dicho cono implica que la hessiana del hamiltoniano respecto del control es semidefinida positiva; en el caso coercivo se obtiene una cota de crecimiento cuadrático sobre esas direcciones. El enfoque se ilustra en un problema lineal–cuadrático con control en un cono poliédrico determinado por tres desigualdades, donde la matriz de gradientes tiene rango dos y la hipótesis de rango estándar falla mientras que la condición de Legendre–Clebsch generalizada sigue siendo válida, proporcionando un criterio más fino en problemas degenerados.

Palabras clave


control óptimo; problemas de Bolza; restricciones mixtas; condiciones de segundo orden; condición de Legendre–Clebsch; hipótesis de rango débil; cono crítico radial.

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Referencias


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DOI: https://doi.org/10.23857/pc.v10i12.10818

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