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Abstract:
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A convex continuous-time maximization problem is formulated and the nec-
essary optimality conditions in the infinite-dimensional case are obtained. As a main tool for
obtaining optimal conditions in this dissertation we use the new theorem of the alternative.
Since there’s no a differentiability assumption, we perform a linearization of the problem
using subdifferentials. It is proved that the multiplier with the objective function won’t be
equal to zero. It was also shown that if the linear and non-linear constraints are separated,
with additional assumptions it can be guaranteed that the multiplier with non-linear constraints
will also be non-zero. In the following, an integral constraint is added to the original convex
problem, so that a Lyapunov-type problem, i.e. an isoperimetric problem, is considered. Lin-
earization of the problem using subdifferentials proved to be a practical way to ignore the lack
of differentiability, so the optimality conditions were derived in a similar way. It is shown that
the obtained results will also be valid for the vector case of the isoperimetric problem.
Additionally, the optimality conditions for the smooth problem were considered. On the
minimization problem, it was shown that the necessary conditions of Karush-Kuhn-Tucker type
will be valid with the additional regularity constraint condition. Also, any point that satisfies
the mentioned optimality conditions will be a global minimum. |