Solution of a specific class of nonlinear fractional optimal control problems including multiple delays

This research provides a new framework based on a hybrid of block‐pulse functions and Legendre polynomials for the numerical examination of a special class of scalar nonlinear fractional optimal control problems involving delay. The concepts of the fractional derivative and the fractional integral are employed in the Caputo sense and the Riemann‐Liouville sense, respectively. In accordance with the notion of the Riemann‐Liouville integral, we derive a new integral operator related to the proposed basis called the operational matrix of fractional integration. By employing two significant operators, namely, the delay operator and the integral operator connected to the hybrid basis, the system dynamics of the primal optimal control problem converts to another system involving algebraic equations. Consequently, the optimal control problem under study is reduced to a static optimization one that is solved by existing well‐established optimization procedures. Some new theoretical results regarding the new basis are obtained. Various kinds of fractional optimal control problems containing delay are examined to measure the accuracy of the new method. The simulation results justify the merits and superiority of the devised procedure over the existing optimization methods in the literature.     

Stabilization of positive coupled differential‐difference equations with unbounded time‐varying delays

This paper researches the static output‐feedback stabilization of single‐input single‐output (SISO) positive coupled differential‐difference equations (CDDEs) with unbounded time‐varying delays. First, a necessary and sufficient condition is provided for the positivity and asymptotical stability of CDDEs with unbounded time‐varying delays. For this type of system, based on the constructed estimates of its solution, a necessary and sufficient condition on asymptotical stability is provided. Then, based on this criterion, for CDDEs with unbounded time‐varying delays, a kind of static output‐feedback controller is designed to ensure the positivity and asymptotical stability of the corresponding closed‐loop systems. It is also worth pointing out that the controller is designed by the linear programming method without parameterization technique. This design approach can also be applied to the static state feedback stabilization problem of CDDEs with unbounded time‐varying delays. Finally, two illustrative examples are given to show the effectiveness of our results.     

Optimal control problems with Atangana‐Baleanu fractional derivative

In this paper, we study fractional‐order optimal control problems (FOCPs) involving the Atangana‐Baleanu fractional derivative. A computational method based on B‐spline polynomials and their operational matrix of Atangana‐Baleanu fractional integration is proposed for the numerical solution of this class of problems. With this numerical technique, the FOCPs are reduced to a system of equations which are solved for the unknown parameters with the help of Mathematica® software. Our results show the applicability and usefulness of the numerical technique.     

Feasibility of a self‐assembling peptide hydrogel scaffold for meniscal defect: An in vivo study in a rabbit model

The inner avascular zone of the meniscus has limited healing capacity as the area is poorly vascularized. Although peptide hydrogels have been reported to regenerate bone and cartilage, their effect on meniscus regeneration remains unknown. We tested whether the self‐assembling peptide hydrogel scaffold KI24RGDS stays in the meniscal lesion and facilitates meniscal repair and regeneration in an induced rabbit meniscal defect model. Full‐thickness (2.0 mm diameter) cylindrical defects were introduced into the inner avascular zones of the anterior portions of the medial menisci of rabbit knees (n = 40). Right knee defects were left empty (control group) while the left knee defects were transplanted with peptide hydrogel (KI24RGDS group). Macroscopic meniscus scores were significantly higher in the KI24RGDS group than in the control group at 2, 4, and 8 weeks after surgery. Histological examinations including quantitative and qualitative scores indicated that compared with the control group, the reparative tissue in the meniscus was significantly enhanced in the KI24RGDS group at 2, 4, 8, and 12 weeks after surgery. Immunohistochemical staining showed that the reparative tissue induced by KI24RGDS at 12 weeks postimplantation was positive for Type I and II collagen. KI24RGDS is highly biocompatible and biodegradable, with strong stiffness, and a three dimensional structure mimicking native extracellular matrix and RGDS sequences that enhance cell adhesion and proliferation. This in vivo study demonstrated that KI24RGDS remained in the meniscal lesion and facilitated the repair and regeneration in a rabbit meniscal defect model.     

Inhibition of Smad3 promotes the healing of rotator cuff injury in a rat model

To investigate the effect of inhibiting transforming growth factor‐β (TGF‐β1)/Smad2/3 signaling on rotator cuff (RC) healing. A bilateral supraspinatus tendon detachment‐repair model of Sprague‐Dawley (SD) rats was utilized. A total of 120 SD rats were randomly assigned to six groups and each group received the subacromial injection of normal saline, empty vectors, or lentiviral vectors containing small interfering RNA against TGF‐β1, Smad2, Smad3 at the bone‐tendon junction. Biomechanical and histological analyses were performed to evaluate bone‐tendon junction healing quality at 8 weeks after repair. Histologically, scar healing was found in all surgical groups. Animals with inhibited Smad3 exhibited better bone‐tendon junction structures with higher density, parallel orientation, and collagen fiber continuity than other surgical group animals. Immunohistochemistry revealed that the protein expression level of collagen I in animals with inhibited Smad3 was more prominent compared with all other surgical groups. Biomechanically, Animals with inhibited Smad3 showed better results in the maximum load at 4, 6, and 8 weeks after surgery compared with other surgical groups. Besides, C3H10T1/2 (Smad3?) cells increased TT‐D6 cell migration and tendon‐associated genes expression (scleraxis, tenascin C, collagen I) in coculture system. We conclude that inhibition of Smad3 promotes RC tendon healing in the rat supraspinatus model.     

Effects of cobalt and chromium ions on glycolytic flux and the stabilization of hypoxia‐inducible factor‐1α in macrophages in vitro

Implant wear and corrosion have been associated with adverse tissue reactions that can lead to implant failure. Wear and corrosion products are therefore of great clinical concern. For example, Co²⁺ and Cr³⁺ originating from CoCrMo‐based implants have been shown to induce a proinflammatory response in macrophages in vitro. Previous studies have also shown that the polarization of macrophages by some proinflammatory stimuli is associated with a hypoxia‐inducible factor‐1α (HIF‐1α)‐dependent metabolic shift from oxidative phosphorylation (OXPHOS) towards glycolysis. However, the potential of Co²⁺ and Cr³⁺ to induce this metabolic shift, which plays a determining role in the proinflammatory response of macrophages, remains largely unexplored. We recently demonstrated that Co²⁺, but not Cr³⁺, increased oxidative stress and decreased OXPHOS in RAW 264.7 murine macrophages. In the present study, we analyzed the effects of Co²⁺ and Cr³⁺ on glycolytic flux and HIF‐1α stabilization in the same experimental model. Cells were exposed to 6 to 24 ppm Co²⁺ or 50 to 250 ppm Cr³⁺. Glycolytic flux was determined by analyzing extracellular flux and lactate production, while HIF‐1α stabilization was analyzed by immunoblotting. Results showed that Co²⁺, and to a lesser extent Cr³⁺, increased glycolytic flux; however, only Co²⁺ acted through HIF‐1α stabilization. Overall, these results, together with our previous results showing that Co²⁺ increases oxidative stress and decreases OXPHOS, suggest that Co²⁺ (but not Cr³⁺) can induce a HIF‐1α‐dependent metabolic shift from OXPHOS towards glycolysis in macrophages. This metabolic shift may play an early and pivotal role in the inflammatory response induced by Co²⁺ in the periprosthetic environment.