Summary: Constrained approximate controllability for linear dynamical systems described by abstract differential equations with unbounded control operator is considered. Using methods of spectral analysis for linear self-adjoint operators and general constrained controllability results given by \textit{N. K. Son} [J. Math. Anal. Appl. 150, 1-19 (1990; Zbl 0715.93004)], necessary and sufficient conditions for the constrained approximate controllability of the piecewise polynomial controls with values in a given cone are formulated and proved. Moreover, as illustrative examples, constrained approximate boundary controllability of one-dimensional distributed parameter dynamical systems described by partial differential equations of parabolic type with Dirichlet and Neumann boundary conditions are investigated. The constrained controllability conditions obtained in the paper represent an extension of the unconstrained controllability results given in [\textit{G. McGlothin}, Int. J. Control 20, 417-432 (1974; Zbl 0286.93022), the author, Controllability of dynamical systems (1991; Zbl 0732.93008)].