Jean-Louis Loday introduced a class of symmetric operads generated by one bilinear operation subject to one relation making each left-normed product of three elements equal to a linear combination of right-normed products: $({{a}_{1}}{{a}_{2}}){{a}_{3}}\,=\,\sum{{{_{\sigma }}_{\in {{s}_{3}}}}\,}{{x}_{\sigma }}\,{{a}_{\sigma \,(1)}}({{a}_{\sigma (2)\,}}{{a}_{\sigma (3)}})$. Such an operad is called a parametrized one-relation operad. For a particular choice of parameters $\{{{x}_{\sigma }}\}$, this operad is said to be regular if each of its components is the regular representation of the symmetric group; equivalently, the corresponding free algebra on a vector space $V$ is, as a graded vector space, isomorphic to the tensor algebra of $V$. We classify, over an algebraically closed field of characteristic zero, all regular parametrized one-relation operads. In fact, we prove that each such operad is isomorphic to one of the following five operads: the left-nilpotent operad defined by the relation $(({{a}_{1}}{{a}_{2}}){{a}_{3}})\,=\,0$, the associative operad, the Leibniz operad, the dual Leibniz (Zinbiel) operad, and the Poisson operad. Our computational methods combine linear algebra over polynomial rings, representation theory of the symmetric group, and Gröbner bases for determinantal ideals and their radicals.

Various algorithms for optimal control require the explicit determination of switching surfaces. However, switching strategies may be very complicated, such that the computation of switching surfaces is quite challenging. General methods are proposed here to compute switching surfaces systematically, based on algebraic computational tools such as triangular decomposition. Our methods are highly complex compared to some widely-used numerical options, but they can be made feasible for realtime applications by moving the computational burden off-line. The tutorial-style presentation is intended to introduce potentially powerful symbolic computation methods to system scientists in particular, and an illustrative example of time-optimal control is given to show the effectiveness and generality of our approach.