Engineers seek optimal solutions when designing dynamic systems but a crucial element is to ensure bounded performance over time. Finding a globally optimal bounded trajectory requires the solution of the ordinary differential equation (ODE) systems in a verified way. To date these methods are only able to address low dimensional problems and for larger systems are unable to prevent gross overestimation of the bounds. In this paper we show how interval contractors can be used to obtain tightly bounded optima. A verified solver constructs tight upper and lower bounds on the dynamic variables using contractors for initial value problems (IVP) for ODEs within a global optimisation method. The solver provides guaranteed bound on the objective function and on the first order sensitivity equations in a branch and bound framework. The method is compared with three previously published methods on three examples from process engineering. (C) 2017 Elsevier Ltd. All rights reserved.