  This paper looks to apply the mathematical framework of the Theory of
Functional Connections to the solution of boundary-value problems arising from
hybrid systems. The Theory of Functional Connections is a technique to derive
constrained expressions which are analytical expressions with embedded
constraints. These expressions are particularly suitable to transform a large
class of constrained optimization problems into unconstrained problems. The
initial and most useful application of this technique is in the solution of
differential equations where the problem can be posed as an unconstrained
optimization problem and solved with simple numerical techniques (i.e.
least-squares).
  A hybrid system is simply a sequence of different differential equations. The
approach developed in this work derives an analytical constrained expression
for the entire range of a hybrid system, enforcing both the boundary conditions
as well as the continuity conditions across the sequence of differential
equations. This reduces the searched solution space of the hybrid system to
only admissible solutions. The transformation allows for a least-squares
solution of the sequence for linear differential equations and a iterative
least-squares solution for nonlinear differential equations.
  Lastly, the general formulation for "n" segments is developed and validation
is provided through numerical tests for three differential equation sequences:
linear/linear, linear/nonlinear, and nonlinear/nonlinear. The accuracy level
obtained are all at machine-error, which is consistent with the accuracy
experienced in past studies on the application of the Theory of Functional
Connections to solve single ordinary differential equations.
