I know of work adding temporal modalities to linear logic. This is quite interesting: a formula (without a modality) is interpreted as resources being available now. The next time modality ($\circ$) is interpreted as resources being available in the next time step. The box modality $\Box$ means that the resources can be consumed at any point in the future, determined by the holder of the resources, whereas $\diamond$ means that the resources can be consumed at any point in time determined by the system. Notice the duality.
Banbara, M., Kang, K.-S., Hirai, T., Tamura, N.: Logic programming in a fragment of intuitionistic temporal linear logic. In: Codognet, P. (ed.) ICLP 2001. LNCS, vol. 2237, pp. 315–330. Springer, Heidelberg (2001)
Hirai, T.: Propositional temporal linear logic and its application to concurrent systems. EICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences (Special Section on Concurrent Systems Technology) E83- A(11), 2219–2227 (2000)
Hirai, T.: Temporal Linear Logic and Its Application. PhD thesis, The Graduate School of Science and Technology, Kobe University, Japan (September 2000).
Kamide, N.: Temporalizing Linear Logic Bulletin of the Section of Logic Volume 36:3/4 (2007), pp. 173–182
There are a few papers adding all sorts of modalities to linear and affine logic:
Kamide, N.: Linear and affine logics with temporal, spatial and epistemic logics. Theoretical Computer Science 252, 165–207 (2006).
Kamide, N: Combining Soft Linear Logic and Spatio-Temporal Operators J Logic Computation (2004) 14 (5): 625-650.
The work on temporal linear logic has been applied in agent-oriented programming and coordination, making essential use of the interpretation of the modalities described above:
Kungas, P.: Temporal linear logic for symbolic agent negotiation. In: Zhang, C., W. Guesgen, H., Yeap, W.-K. (eds.) PRICAI 2004. LNCS, vol. 3157, pp. 23–32. Springer, Heidelberg (2004)
Pham, D.Q., Harland, J., Winikoff, M.: Modelling agent’s choices in temporal linear logic. In: Baldoni, M., Son, T.C., van Riemsdijk, M.B., Winikoff, M. (eds.) DALT 2007. LNCS, vol. 4897, pp. 140–157. Springer, Heidelberg (2008)
Dave Clarke. Coordination: Reo, Nets and Logic. FMCO proceedings, LNCS, vol. 5382. (2008)