Many practical optimal control problems include discrete decisions. These may be either time–independent parameters or time–dependent control functions as gears or valves that can only take discrete values at any given time. While great progress has been achieved in the solution of optimization problems involving integer variables, in particular mixed–integer linear programs, as well as in cont...

The shoot2.0 package implements an indirect shooting method for optimal control problems. It is speci cally designed to handle control discontinuities, with an automatic switching detection that requires no assumptions concerning the number of switchings. Special care is also devoted to the computation of the Jacobian matrix of the shooting function, using the variational system instead of clas...

where 0 < αi < βi < 1, 0 < ξi < ηi < 1, γi > 0, ρi > 0, i = 1, 2, . . . ,n, λ > 0, μ > 0. J = [0, 1], J∗ = J\{t1, t2, . . . , tp}, 0 = t0 < t1 < t2 < · · · < tp < tp+1 = 1, u(tk) = u(t+ k ) – u(t– k ) denotes the jump of u(t) at t = tk , u(t+ k ) and u(t – k ) represent the right and left limits of u(t) at t = tk , respectively, u′(tk) has a similar meaning for u′(t). In fact, multi-strip condi...

A new adaptive L₁/₂ shooting regularization method for variable selection based on the Cox's proportional hazards mode being proposed. This adaptive L₁/₂ shooting algorithm can be easily obtained by the optimization of a reweighed iterative series of L₁ penalties and a shooting strategy of L₁/₂ penalty. Simulation results based on high dimensional artificial data show that the adaptive L₁/₂ sho...

* Correspondence: [email protected] Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, PR China Abstract In this article, we consider the uniqueness of positive radial solutions to the Dirichlet boundary value problem u + f (|x|, u) + g(|x|)x · ∇u = 0, x ∈ , u = 0, x ∈ ∂ , where Ω denotes an annulus in R (n ≥ 3). The uniqueness criterion is established by applyi...

We study both existence and the exact number of positive solutions of the problem . All righ . Iturria ðPkÞ k ju0j u0 0 þ f ðuÞ 1⁄4 0 in ð0;1Þ; uð0Þ 1⁄4 uð1Þ 1⁄4 0; <: where k is a positive parameter, p > 1, the nonlinearity f is positive in (0,1), and f ð0Þ 1⁄4 f ð1Þ 1⁄4 0. Assuming that f satisfies the condition lims!1 f ðsÞ ð1 sÞ 1⁄4 x > 0 where h 2 ð0; p 1Þ, we study its behavior near zero,...