Assessment and Future Directions of Nonlinear Model Predictive ...

40 É. Gyurkovics and A.M. ElaiwWe might formulate this assumptions in part with respect to the approximatediscrete-time model as it was done e.g. in [5] and [15]. However, it turns outthat in several cases the verification of some conditions is much more tractablefor the original model than the approximate one. For this reason, we formulatethe assumptions with respect to the exact model (and to the applied numericalapproximation method). For the design parameters l and g, we shall make thefollowing assumption.Assumption 1. (i) g : R n → R is continuous, positive definite, radially unboundedand Lipschitz continuos in any compact set.(ii) l is continuous with respect to x and u and Lipschitz continuous with respectto x in any compact set.(iii) There exist such class-K ∞ functions ϕ 1 , ϕ 1 ,ϕ 2 and ϕ 2 thatϕ 1 (‖x‖)+ϕ 1 (‖u‖) ≤ l(x, u) ≤ ϕ 2 (‖x‖)+ϕ 2 (‖u‖), (9)holds for all x ∈X and u ∈ U.Remark 2. The lower bound in (9) can be substituted by different conditions:e.g. ϕ 1 may be omitted, if U is compact. If the stage cost for the discrete-timeoptimization problem is directly given, other conditions ensuring the existenceand uniform boundedness of the optimal control sequence can be imposed, aswell (see e.g. [10], [15] and [20] ). However, having a K ∞ lower estimation withrespect to ‖x‖ is important in the considerations of the present paper.The applied numerical approximation scheme has to ensure the closeness of theexact and the approximate models in the following sense.Assumption 2. For any given ∆ ′ > 0and∆ ′′ > 0thereexistsah ∗ 0 > 0suchthat(i) FT,h A (0, 0) = 0, lA T,h (0, 0) = 0, lA T,h (x, u) > 0, x ≠0,F T,h A and lA T,h arecontinuous in both variables uniformly in h ∈ (0,h ∗ 0 ], and they preserve theLischitz continuity of the exact models, uniformly in h;(ii) there exists a γ ∈Ksuch that‖F E T (x, u) − F A T,h(x, u)‖ ≤Tγ(h),for all x ∈B ∆ ′,allu ∈ U ∆′′ ,andh ∈ (0,h ∗ 0 ].‖l E T (x, u) − l A T,h(x, u)‖ ≤Tγ(h),Remark 3. We note that Assumption A2 depends on the numerical approximationmethod, and it can be proven for reasonable discretization formulas.Definition 1. System (3) is asymptotically controllable from a compact set Ωto the origin, if there exist a β(., .) ∈KLand a continuous, positive and nondecreasingfunction σ(.) such that for all x ∈ Ω there exists a control sequenceu(x), u k (x) ∈ U, such that ‖u k (x)‖ ≤σ(‖x‖), and the corresponding solutionφ E of (3) satisfies the inequality∥ φEk (x, u(x)) ∥ ∥ ≤ β(‖x‖ ,kT), k ∈ N.