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][1]. 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][2]. 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][3]. PhD thesis, The Graduate School of Science and Technology, Kobe University, Japan (September 2000).

There is also at least one paper adding all sorts of modalities to linear and affine logic:

* Kamide, N.: [Linear and affine logics with temporal, spatial and epistemic logics][4]. Theoretical Computer Science 252, 165–207 (2006).

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][5]. 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][6]. 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][7]. 
FMCO proceedings, LNCS, vol.  5382. (2008)


  [1]: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.77.6666&rep=rep1&type=pdf
  [2]: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.2107
  [3]: http://kaminari.scitec.kobe-u.ac.jp/pub/hiraiPhD00.pdf
  [4]: http://portal.acm.org/citation.cfm?id=1143365
  [5]: http://www.springerlink.com/index/r87l9g0mw7mvknr6.pdf
  [6]: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.64.9407&rep=rep1&type=pdf
  [7]: https://lirias.kuleuven.be/handle/123456789/217971